Allow to union MultiIndex with empty RangeIndex

pandas/core/indexes/base.py

@@ -2923,8 +2923,10 @@ def union(self, other, sort=None):
         other, result_name = self._convert_can_do_setop(other)
 
         if not is_dtype_equal(self.dtype, other.dtype):
-            if isinstance(self, ABCMultiIndex) and not is_object_dtype(
-                unpack_nested_dtype(other)
+            if (
+                isinstance(self, ABCMultiIndex)
+                and not is_object_dtype(unpack_nested_dtype(other))
+                and len(other) > 0
             ):
                 raise NotImplementedError(
                     "Can only union MultiIndex with MultiIndex or Index of tuples, "

pandas/core/indexes/multi.py

@@ -3597,7 +3597,7 @@ def _get_reconciled_name_object(self, other) -> MultiIndex:
     def _maybe_match_names(self, other):
         """
         Try to find common names to attach to the result of an operation between
-        a and b.  Return a consensus list of names if they match at least partly
+        a and b. Return a consensus list of names if they match at least partly
         or list of None if they have completely different names.
         """
         if len(self.names) != len(other.names):

pandas/tests/indexes/multi/test_setops.py

@@ -414,6 +414,14 @@ def test_union_empty_self_different_names():
     tm.assert_index_equal(result, expected)
 
 
+def test_union_multiindex_empty_rangeindex():
+    # GH#41234
+    mi = MultiIndex.from_arrays([[1, 2], [3, 4]], names=["a", "b"])
+    ri = pd.RangeIndex(0)
+    tm.assert_index_equal(mi, mi.union(ri), check_names=False)
+    tm.assert_index_equal(mi, ri.union(mi), check_names=False)
+
+
 @pytest.mark.parametrize(
     "method", ["union", "intersection", "difference", "symmetric_difference"]
 )

pandas/tests/reshape/concat/test_concat.py

@@ -627,3 +627,14 @@ def test_concat_null_object_with_dti():
         index=exp_index,
     )
     tm.assert_frame_equal(result, expected)
+
+
+def test_concat_multiindex_with_empty_rangeindex():
+    # GH#41234
+    mi = MultiIndex.from_tuples([("B", 1), ("C", 1)])
+    df1 = DataFrame([[1, 2]], columns=mi)
+    df2 = DataFrame(index=[1], columns=pd.RangeIndex(0))
+
+    result = concat([df1, df2])
+    expected = DataFrame([[1, 2], [np.nan, np.nan]], columns=mi)
+    tm.assert_frame_equal(result, expected)