REGR: MultiIndex.join does not work for ea dtypes (pandas-dev#49284)

* REGR: MultiIndex.join does not work for ea dtypes

* Update base.py

Author: phofl

doc/source/whatsnew/v1.5.2.rst

@@ -13,6 +13,7 @@ including other versions of pandas.
 
 Fixed regressions
 ~~~~~~~~~~~~~~~~~
+- Fixed regression in :meth:`MultiIndex.join` for extension array dtypes (:issue:`49277`)
 - Fixed regression in :meth:`Series.replace` raising ``RecursionError`` with numeric dtype and when specifying ``value=None`` (:issue:`45725`)
 - Fixed regression in :meth:`DataFrame.plot` preventing :class:`~matplotlib.colors.Colormap` instance
   from being passed using the ``colormap`` argument if Matplotlib 3.6+ is used (:issue:`49374`)

pandas/core/indexes/base.py

@@ -4456,8 +4456,10 @@ def join(
             return self._join_non_unique(other, how=how)
         elif not self.is_unique or not other.is_unique:
             if self.is_monotonic_increasing and other.is_monotonic_increasing:
-                if self._can_use_libjoin:
+                if not is_interval_dtype(self.dtype):
                     # otherwise we will fall through to _join_via_get_indexer
+                    # GH#39133
+                    # go through object dtype for ea till engine is supported properly
                     return self._join_monotonic(other, how=how)
             else:
                 return self._join_non_unique(other, how=how)
@@ -4832,7 +4834,7 @@ def _wrap_joined_index(self: _IndexT, joined: ArrayLike, other: _IndexT) -> _Ind
             return self._constructor(joined, name=name)  # type: ignore[return-value]
         else:
             name = get_op_result_name(self, other)
-            return self._constructor._with_infer(joined, name=name)
+            return self._constructor._with_infer(joined, name=name, dtype=self.dtype)
 
     @cache_readonly
     def _can_use_libjoin(self) -> bool:

pandas/tests/indexes/multi/test_join.py

@@ -6,6 +6,8 @@
     Index,
     Interval,
     MultiIndex,
+    Series,
+    StringDtype,
 )
 import pandas._testing as tm
 
@@ -161,6 +163,52 @@ def test_join_overlapping_interval_level():
     tm.assert_index_equal(result, expected)
 
 
+def test_join_midx_ea():
+    # GH#49277
+    midx = MultiIndex.from_arrays(
+        [Series([1, 1, 3], dtype="Int64"), Series([1, 2, 3], dtype="Int64")],
+        names=["a", "b"],
+    )
+    midx2 = MultiIndex.from_arrays(
+        [Series([1], dtype="Int64"), Series([3], dtype="Int64")], names=["a", "c"]
+    )
+    result = midx.join(midx2, how="inner")
+    expected = MultiIndex.from_arrays(
+        [
+            Series([1, 1], dtype="Int64"),
+            Series([1, 2], dtype="Int64"),
+            Series([3, 3], dtype="Int64"),
+        ],
+        names=["a", "b", "c"],
+    )
+    tm.assert_index_equal(result, expected)
+
+
+def test_join_midx_string():
+    # GH#49277
+    midx = MultiIndex.from_arrays(
+        [
+            Series(["a", "a", "c"], dtype=StringDtype()),
+            Series(["a", "b", "c"], dtype=StringDtype()),
+        ],
+        names=["a", "b"],
+    )
+    midx2 = MultiIndex.from_arrays(
+        [Series(["a"], dtype=StringDtype()), Series(["c"], dtype=StringDtype())],
+        names=["a", "c"],
+    )
+    result = midx.join(midx2, how="inner")
+    expected = MultiIndex.from_arrays(
+        [
+            Series(["a", "a"], dtype=StringDtype()),
+            Series(["a", "b"], dtype=StringDtype()),
+            Series(["c", "c"], dtype=StringDtype()),
+        ],
+        names=["a", "b", "c"],
+    )
+    tm.assert_index_equal(result, expected)
+
+
 def test_join_multi_with_nan():
     # GH29252
     df1 = DataFrame(