BUG/PERF: use lexsort_indexer in MultiIndex.argsort

pandas/core/indexes/multi.py

@@ -1940,7 +1940,7 @@ def _lexsort_depth(self) -> int:
             return self.sortorder
         return _lexsort_depth(self.codes, self.nlevels)
 
-    def _sort_levels_monotonic(self, raise_if_unsortable: bool = False) -> MultiIndex:
+    def _sort_levels_monotonic(self, raise_if_incomparable: bool = False) -> MultiIndex:
         """
         This is an *internal* function.
 
@@ -1987,7 +1987,7 @@ def _sort_levels_monotonic(self, raise_if_unsortable: bool = False) -> MultiInde
                     # indexer to reorder the levels
                     indexer = lev.argsort()
                 except TypeError:
-                    if raise_if_unsortable:
+                    if raise_if_incomparable:
                         raise
                 else:
                     lev = lev.take(indexer)
@@ -2235,7 +2235,7 @@ def append(self, other):
     def argsort(self, *args, **kwargs) -> npt.NDArray[np.intp]:
         if len(args) == 0 and len(kwargs) == 0:
             # lexsort is significantly faster than self._values.argsort()
-            target = self._sort_levels_monotonic(raise_if_unsortable=True)
+            target = self._sort_levels_monotonic(raise_if_incomparable=True)
             return lexsort_indexer(target._get_codes_for_sorting())
         return self._values.argsort(*args, **kwargs)
 

pandas/tests/indexes/multi/test_sorting.py

@@ -14,6 +14,7 @@
     Index,
     MultiIndex,
     RangeIndex,
+    Timestamp,
 )
 import pandas._testing as tm
 from pandas.core.indexes.frozen import FrozenList
@@ -289,3 +290,20 @@ def test_sort_values_nan():
         levels=[["A", "B", "C"], ["D"]], codes=[[0, 1, 2], [-1, -1, 0]]
     )
     tm.assert_index_equal(result, expected)
+
+
+def test_sort_values_monotonic_incomparable():
+    mi = MultiIndex.from_arrays(
+        [
+            [1, Timestamp("2000-01-01")],
+            [3, 4],
+        ]
+    )
+
+    match = "'<' not supported between instances of 'Timestamp' and 'int'"
+    with pytest.raises(TypeError, match=match):
+        mi._sort_levels_monotonic(raise_if_incomparable=True)
+
+    result = mi._sort_levels_monotonic()
+    expected = mi
+    tm.assert_index_equal(result, expected)