diff --git a/CHANGELOG.md b/CHANGELOG.md
index a60ac7a..bab9997 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -2,6 +2,9 @@
 
 ## 1.5.0
 
+  * Make the `Eq` and `Ord` instances for key-value queues sensible.
+    Document the nondeterministic nature of these queues.
+
   * Make mapping and traversal functions force the full data structure spine.
     This should make performance more predictable, and removes the last
     remaining reasons to use the `seqSpine` functions. As these are no longer
diff --git a/src/BinomialQueue/Max.hs b/src/BinomialQueue/Max.hs
index ebb2afd..397fcf3 100644
--- a/src/BinomialQueue/Max.hs
+++ b/src/BinomialQueue/Max.hs
@@ -169,7 +169,7 @@ take :: Ord a => Int -> MaxQueue a -> [a]
 take n = List.take n . toDescList
 
 -- | \(O(k \log n)\)/. 'drop' @k@, applied to a queue @queue@, returns @queue@ with the greatest @k@ elements deleted,
--- or an empty queue if @k >= size 'queue'@.
+-- or an empty queue if @k >= 'size' queue@.
 drop :: Ord a => Int -> MaxQueue a -> MaxQueue a
 drop n (MaxQueue queue) = MaxQueue (MinQ.drop n queue)
 
diff --git a/src/BinomialQueue/Min.hs b/src/BinomialQueue/Min.hs
index 3fc0a86..dfd5fa9 100644
--- a/src/BinomialQueue/Min.hs
+++ b/src/BinomialQueue/Min.hs
@@ -162,7 +162,7 @@ take :: Ord a => Int -> MinQueue a -> [a]
 take n = List.take n . toAscList
 
 -- | \(O(k \log n)\)/. 'drop' @k@, applied to a queue @queue@, returns @queue@ with the smallest @k@ elements deleted,
--- or an empty queue if @k >= size 'queue'@.
+-- or an empty queue if @k >= 'size' queue@.
 drop :: Ord a => Int -> MinQueue a -> MinQueue a
 drop n queue = n `seq` case minView queue of
   Just (_, queue')
diff --git a/src/Data/PQueue/Internals.hs b/src/Data/PQueue/Internals.hs
index cd6f0d7..d40e18b 100644
--- a/src/Data/PQueue/Internals.hs
+++ b/src/Data/PQueue/Internals.hs
@@ -198,8 +198,9 @@ mapEither f (MinQueue _ x ts)
       Left y -> (fromBare (BQ.insert y l), fromBare r)
       Right z -> (fromBare l, fromBare (BQ.insert z r))
 
--- | \(O(n)\). Assumes that the function it is given is monotonic, and applies this function to every element of the priority queue,
--- as in 'fmap'. If it is not, the result is undefined.
+-- | \(O(n)\). Assumes that the function it is given is monotonic, and applies
+-- this function to every element of the priority queue, as in 'fmap'. If it is
+-- not, the result is undefined.
 mapMonotonic :: (a -> b) -> MinQueue a -> MinQueue b
 mapMonotonic = mapU
 
@@ -274,7 +275,7 @@ insertMinQ' x (MinQueue n x' f) = MinQueue (n + 1) x (BQ.insertMinQ' x' f)
 
 -- | @insertMaxQ' x h@ assumes that @x@ compares as greater
 -- than or equal to every element of @h@. It also assumes,
--- and preserves, an extra invariant. See 'insertMax'' for details.
+-- and preserves, an extra invariant. See 'BQ.insertMax'' for details.
 -- tldr: this function can be used safely to build a queue from an
 -- ascending list/array/whatever, but that's about it.
 insertMaxQ' :: a -> MinQueue a -> MinQueue a
@@ -289,6 +290,9 @@ fromList :: Ord a => [a] -> MinQueue a
 -- comparison per element.
 fromList xs = fromBare (BQ.fromList xs)
 
+-- | \(O(n)\). Assumes that the function it is given is monotonic, and applies
+-- this function to every element of the priority queue, as in 'fmap'. If it is
+-- not, the result is undefined.
 mapU :: (a -> b) -> MinQueue a -> MinQueue b
 mapU _ Empty = Empty
 mapU f (MinQueue n x ts) = MinQueue n (f x) (BQ.mapU f ts)
diff --git a/src/Data/PQueue/Min.hs b/src/Data/PQueue/Min.hs
index 21733ba..ee2229e 100644
--- a/src/Data/PQueue/Min.hs
+++ b/src/Data/PQueue/Min.hs
@@ -128,7 +128,7 @@ pattern Empty = Internals.Empty
 infixr 5 :<
 
 -- | A bidirectional pattern synonym for working with the minimum view of a
--- 'MinPQueue'.  Using @:<@ to construct a queue performs an insertion in
+-- 'Prio.MinPQueue'.  Using @:<@ to construct a queue performs an insertion in
 -- \(O(1)\) amortized time. When matching on @a :< q@, forcing @q@ takes
 -- \(O(\log n)\) time.
 --
@@ -204,7 +204,7 @@ take :: Ord a => Int -> MinQueue a -> [a]
 take n = List.take n . toAscList
 
 -- | \(O(k \log n)\)/. 'drop' @k@, applied to a queue @queue@, returns @queue@ with the smallest @k@ elements deleted,
--- or an empty queue if @k >= size 'queue'@.
+-- or an empty queue if @k >= 'size' queue@.
 drop :: Ord a => Int -> MinQueue a -> MinQueue a
 drop n queue = n `seq` case minView queue of
   Just (_, queue')
diff --git a/src/Data/PQueue/Prio/Internals.hs b/src/Data/PQueue/Prio/Internals.hs
index 38edea7..77ad1a3 100644
--- a/src/Data/PQueue/Prio/Internals.hs
+++ b/src/Data/PQueue/Prio/Internals.hs
@@ -52,6 +52,7 @@ module Data.PQueue.Prio.Internals (
 
 import Control.Applicative (liftA2, liftA3)
 import Control.DeepSeq (NFData(rnf), deepseq)
+import Data.Function (on)
 import Data.Functor.Identity (Identity(Identity, runIdentity))
 import qualified Data.List as List
 import Data.PQueue.Internals.Foldable
@@ -210,37 +211,18 @@ instance IFoldMap t => IFoldMap (BinomForest t) where
   foldMapWithKey_ f (Skip ts) = foldMapWithKey_ f ts
   foldMapWithKey_ f (Cons t ts) = foldMapWithKey_ f t `mappend` foldMapWithKey_ f ts
 
-instance (Ord k, Eq a) => Eq (MinPQueue k a) where
-  MinPQ n1 k1 a1 ts1 == MinPQ n2 k2 a2 ts2 =
-    n1 == n2 && eqExtract k1 a1 ts1 k2 a2 ts2
-  Empty == Empty = True
-  _     == _     = False
-
-eqExtract :: (Ord k, Eq a) => k -> a -> BinomForest rk k a -> k -> a -> BinomForest rk k a -> Bool
-eqExtract k10 a10 ts10 k20 a20 ts20 =
-  k10 == k20 && a10 == a20 &&
-  case (extract ts10, extract ts20) of
-    (Yes (Extract k1 a1 _ ts1'), Yes (Extract k2 a2 _ ts2'))
-             -> eqExtract k1 a1 ts1' k2 a2 ts2'
-    (No, No) -> True
-    _        -> False
+-- | @ (==) = (==) ``on`` 'List.sort' . 'toAscList' @
+instance (Ord k, Ord a) => Eq (MinPQueue k a) where
+  (==) = (==) `on` toFullySortedList
 
+toFullySortedList :: (Ord k, Ord a) => MinPQueue k a -> [(k, a)]
+-- We break up the list to avoid lots of redundant key comparisons
+-- in sorting.
+toFullySortedList = List.concatMap (List.sortBy (compare `on` snd)) . List.groupBy ((==) `on` fst) . toAscList
+
+-- | @ compare = compare ``on`` 'List.sort' . 'toAscList' @
 instance (Ord k, Ord a) => Ord (MinPQueue k a) where
-  MinPQ _n1 k10 a10 ts10 `compare` MinPQ _n2 k20 a20 ts20 =
-    cmpExtract k10 a10 ts10 k20 a20 ts20
-  Empty `compare` Empty   = EQ
-  Empty `compare` MinPQ{} = LT
-  MinPQ{} `compare` Empty = GT
-
-cmpExtract :: (Ord k, Ord a) => k -> a -> BinomForest rk k a -> k -> a -> BinomForest rk k a -> Ordering
-cmpExtract k10 a10 ts10 k20 a20 ts20 =
-  k10 `compare` k20 <> a10 `compare` a20 <>
-  case (extract ts10, extract ts20) of
-    (Yes (Extract k1 a1 _ ts1'), Yes (Extract k2 a2 _ ts2'))
-                -> cmpExtract k1 a1 ts1' k2 a2 ts2'
-    (No, Yes{}) -> LT
-    (Yes{}, No) -> GT
-    (No, No)    -> EQ
+  compare = compare `on` toFullySortedList
 
 -- | \(O(1)\). Returns the empty priority queue.
 empty :: MinPQueue k a
@@ -343,9 +325,11 @@ minViewWithKey (MinPQ n k a ts) = Just ((k, a), extractHeap n ts)
 mapWithKey :: (k -> a -> b) -> MinPQueue k a -> MinPQueue k b
 mapWithKey f = runIdentity . traverseWithKeyU (Identity .: f)
 
--- | \(O(n)\). @'mapKeysMonotonic' f q == 'mapKeys' f q@, but only works when @f@ is strictly
--- monotonic. /The precondition is not checked./ This function has better performance than
--- 'mapKeys'.
+-- | \(O(n)\). @'mapKeysMonotonic' f q == 'Data.PQueue.Prio.Min.mapKeys' f q@,
+-- but only works when @f@ is strictly monotonic.
+-- /The precondition is not checked./
+-- This function has better performance than 'Data.PQueue.Prio.Min.mapKeys'.
+
 mapKeysMonotonic :: (k -> k') -> MinPQueue k a -> MinPQueue k' a
 mapKeysMonotonic _ Empty = Empty
 mapKeysMonotonic f (MinPQ n k a ts) = MinPQ n (f k) a (mapKeysMonoF f (const Zero) ts)
diff --git a/src/Data/PQueue/Prio/Max.hs b/src/Data/PQueue/Prio/Max.hs
index 1bc39d6..66c2545 100644
--- a/src/Data/PQueue/Prio/Max.hs
+++ b/src/Data/PQueue/Prio/Max.hs
@@ -21,6 +21,9 @@
 -- are no guarantees about the relative order in which @k1@, @k2@, and their associated
 -- elements are returned. (Unlike Data.Map, we allow multiple elements with the
 -- same key.)
+-- The order in which key-value pairs with the same key are extracted, folded,
+-- or traversed is explicitly unspecified. From a semantic standpoint, it is
+-- simplest to imagine that these operations are nondeterministic.
 --
 -- This implementation offers a number of methods of the form @xxxU@, where @U@ stands for
 -- unordered. No guarantees whatsoever are made on the execution or traversal order of
diff --git a/src/Data/PQueue/Prio/Max/Internals.hs b/src/Data/PQueue/Prio/Max/Internals.hs
index 6bfa5ea..3c285f7 100644
--- a/src/Data/PQueue/Prio/Max/Internals.hs
+++ b/src/Data/PQueue/Prio/Max/Internals.hs
@@ -129,6 +129,8 @@ import Text.Read (Lexeme(Ident), lexP, parens, prec,
 import Data.Functor.WithIndex
 import Data.Foldable.WithIndex
 import Data.Traversable.WithIndex
+import qualified Data.List as List
+import Data.Function (on)
 
 #ifndef __GLASGOW_HASKELL__
 build :: ((a -> [a] -> [a]) -> [a] -> [a]) -> [a]
@@ -138,11 +140,27 @@ build f = f (:) []
 -- | A priority queue where values of type @a@ are annotated with keys of type @k@.
 -- The queue supports extracting the element with maximum key.
 newtype MaxPQueue k a = MaxPQ (MinPQueue (Down k) a)
-# if __GLASGOW_HASKELL__
-  deriving (Eq, Ord, Data)
-# else
-  deriving (Eq, Ord)
-# endif
+#ifdef __GLASGOW_HASKELL__
+  deriving (Data)
+#endif
+
+-- | @ (==) = (==) ``on`` 'List.sort' . 'List.map' 'Data.Ord.Down' . 'toDescList' @
+instance (Ord k, Ord a) => Eq (MaxPQueue k a) where
+-- We define an instance rather than deriving one because old Haddock versions
+-- choked on documentation in deriving clauses. *sigh*. That's fixed with GHC
+-- 8.2.
+  MaxPQ p == MaxPQ q = p == q
+
+-- | @ compare = compare ``on`` 'List.sort' . 'List.map' 'Data.Ord.Down' . 'toDescList' @
+instance (Ord k, Ord a) => Ord (MaxPQueue k a) where
+  compare = compare `on` toFullySortedList
+
+toFullySortedList :: (Ord k, Ord a) => MaxPQueue k a -> [Down (k, a)]
+-- Gosh, this is a mess, but it works.
+toFullySortedList (MaxPQ q) =
+  List.concatMap (List.sortBy (compare `on` Down . snd . unDown)) .
+  List.groupBy ((==) `on` fst . unDown) .
+  List.map (\(Down k, v) -> Down (k, v)) . Q.toAscList $ q
 
 instance (NFData k, NFData a) => NFData (MaxPQueue k a) where
   rnf (MaxPQ q) = rnf q
diff --git a/src/Data/PQueue/Prio/Min.hs b/src/Data/PQueue/Prio/Min.hs
index 2a39a90..4ae2110 100644
--- a/src/Data/PQueue/Prio/Min.hs
+++ b/src/Data/PQueue/Prio/Min.hs
@@ -26,6 +26,10 @@
 -- elements are returned. (Unlike Data.Map, we allow multiple elements with the
 -- same key.)
 --
+-- The order in which key-value pairs with the same key are extracted, folded,
+-- or traversed is explicitly unspecified. From a semantic standpoint, it is
+-- simplest to imagine that these operations are nondeterministic.
+--
 -- This implementation offers a number of methods of the form @xxxU@, where @U@ stands for
 -- unordered. No guarantees whatsoever are made on the execution or traversal order of
 -- these functions.
diff --git a/tests/PQueueTests.hs b/tests/PQueueTests.hs
index 18807d2..c854fe5 100644
--- a/tests/PQueueTests.hs
+++ b/tests/PQueueTests.hs
@@ -53,8 +53,8 @@ main = defaultMain $ testGroup "pqueue"
     , testProperty "foldlU" $ \xs -> Min.foldlU (+) 0 (Min.fromList xs) === sum xs
     , testProperty "foldlU'" $ \xs -> Min.foldlU' (+) 0 (Min.fromList xs) === sum xs
     , testProperty "toListU" $ \xs -> List.sort (Min.toListU (Min.fromList xs)) === List.sort xs
-    , testProperty "==" $ \(xs :: [(Int, ())]) ys -> ((==) `on` Min.fromList) xs ys === ((==) `on` List.sort) xs ys
-    , testProperty "compare" $ \(xs :: [(Int, ())]) ys -> (compare `on` Min.fromList) xs ys === (compare `on` List.sort) xs ys
+    , testProperty "==" $ \(xs :: [(Int, Int)]) ys -> ((==) `on` Min.fromList) xs ys === ((==) `on` List.sort) xs ys
+    , testProperty "compare" $ \(xs :: [(Int, Int)]) ys -> (compare `on` Min.fromList) xs ys === (compare `on` List.sort) xs ys
     ]
   , testGroup "Data.PQueue.Max"
     [ testProperty "size" $ \xs -> Max.size (Max.fromList xs) === length xs
@@ -87,8 +87,8 @@ main = defaultMain $ testGroup "pqueue"
     , testProperty "foldlU" $ \xs -> Max.foldlU (+) 0 (Max.fromList xs) === sum xs
     , testProperty "foldlU'" $ \xs -> Max.foldlU' (+) 0 (Max.fromList xs) === sum xs
     , testProperty "toListU" $ \xs -> List.sort (Max.toListU (Max.fromList xs)) === List.sort xs
-    , testProperty "==" $ \(xs :: [(Int, ())]) ys -> ((==) `on` Max.fromList) xs ys === ((==) `on` List.sort) xs ys
-    , testProperty "compare" $ \(xs :: [(Int, ())]) ys -> (compare `on` Max.fromList) xs ys === (compare `on` (List.sort . List.map Down)) xs ys
+    , testProperty "==" $ \(xs :: [(Int, Int)]) ys -> ((==) `on` Max.fromList) xs ys === ((==) `on` List.sort) xs ys
+    , testProperty "compare" $ \(xs :: [(Int, Int)]) ys -> (compare `on` Max.fromList) xs ys === (compare `on` (List.sort . List.map Down)) xs ys
     ]
   , testGroup "Data.PQueue.Prio.Min"
     [ testProperty "size" $ \xs -> PMin.size (PMin.fromList xs) === length xs
@@ -187,7 +187,7 @@ main = defaultMain $ testGroup "pqueue"
     , testProperty "traverseU" $
       \(Fn (f :: () -> Maybe ())) (xs :: [(Int, ())]) -> PMax.traverseU f (PMax.fromList xs) === fmap PMax.fromList (traverse (\(k, x) -> fmap (k,) (f x)) xs)
     , testProperty "toListU" $ \xs -> List.sort (PMax.toListU (PMax.fromList xs)) === List.sort xs
-    , testProperty "==" $ \(xs :: [(Int, ())]) ys -> ((==) `on` PMax.fromList) xs ys === ((==) `on` List.sort) xs ys
-    , testProperty "compare" $ \(xs :: [(Int, ())]) ys -> (compare `on` PMax.fromList) xs ys === (compare `on` (List.sort . List.map Down)) xs ys
+    , testProperty "==" $ \(xs :: [(Int, Int)]) ys -> ((==) `on` PMax.fromList) xs ys === ((==) `on` List.sort) xs ys
+    , testProperty "compare" $ \(xs :: [(Int, Int)]) ys -> (compare `on` PMax.fromList) xs ys === (compare `on` (List.sort . List.map Down)) xs ys
     ]
   ]