BUG: MultiIndex.equals returning incorrectly True when Indexes contains NaN

doc/source/whatsnew/v1.3.0.rst

@@ -232,7 +232,7 @@ MultiIndex
 ^^^^^^^^^^
 
 - Bug in :meth:`DataFrame.drop` raising ``TypeError`` when :class:`MultiIndex` is non-unique and no level is provided (:issue:`36293`)
--
+- Bug in :meth:`MultiIndex.equals` incorrectly returning ``True`` when :class:`MultiIndex` containing ``NaN`` even when they are differntly ordered (:issue:`38439`)
 
 I/O
 ^^^

pandas/core/indexes/multi.py

@@ -3454,13 +3454,17 @@ def equals(self, other: object) -> bool:
 
         for i in range(self.nlevels):
             self_codes = self.codes[i]
-            self_codes = self_codes[self_codes != -1]
+            other_codes = other.codes[i]
+            self_mask = self_codes == -1
+            other_mask = other_codes == -1
+            if not np.array_equal(self_mask, other_mask):
+                return False
+            self_codes = self_codes[~self_mask]
             self_values = algos.take_nd(
                 np.asarray(self.levels[i]._values), self_codes, allow_fill=False
             )
 
-            other_codes = other.codes[i]
-            other_codes = other_codes[other_codes != -1]
+            other_codes = other_codes[~other_mask]
             other_values = algos.take_nd(
                 np.asarray(other.levels[i]._values), other_codes, allow_fill=False
             )

pandas/tests/indexes/multi/test_equivalence.py

@@ -209,6 +209,16 @@ def test_equals_missing_values():
     assert not result
 
 
+def test_equals_missing_values_differently_sorted():
+    # GH#38439
+    mi1 = pd.MultiIndex.from_tuples([(81.0, np.nan), (np.nan, np.nan)])
+    mi2 = pd.MultiIndex.from_tuples([(np.nan, np.nan), (81.0, np.nan)])
+    assert not mi1.equals(mi2)
+
+    mi2 = pd.MultiIndex.from_tuples([(81.0, np.nan), (np.nan, np.nan)])
+    assert mi1.equals(mi2)
+
+
 def test_is_():
     mi = MultiIndex.from_tuples(zip(range(10), range(10)))
     assert mi.is_(mi)