BUG: get_indexer rountripping through string dtype

doc/source/whatsnew/v2.3.0.rst

@@ -118,7 +118,7 @@ Interval
 
 Indexing
 ^^^^^^^^
--
+- Fixed bug in :meth:`Index.get_indexer` round-tripping through string dtype when ``infer_string`` is enabled (:issue:`55834`)
 -
 
 Missing

pandas/core/indexes/base.py

@@ -6556,7 +6556,16 @@ def _maybe_cast_listlike_indexer(self, target) -> Index:
         """
         Analogue to maybe_cast_indexer for get_indexer instead of get_loc.
         """
-        return ensure_index(target)
+        target_index = ensure_index(target)
+        if (
+            not hasattr(target, "dtype")
+            and self.dtype == object
+            and target_index.dtype == "string"
+        ):
+            # If we started with a list-like, avoid inference to string dtype if self
+            # is object dtype (coercing to string dtype will alter the missing values)
+            target_index = Index(target, dtype=self.dtype)
+        return target_index
 
     @final
     def _validate_indexer(

pandas/tests/indexes/object/test_indexing.py

@@ -62,6 +62,15 @@ def test_get_indexer_with_NA_values(
         expected = np.array([0, 1, -1], dtype=np.intp)
         tm.assert_numpy_array_equal(result, expected)
 
+    def test_get_indexer_infer_string_missing_values(self):
+        # ensure the passed list is not cast to string but to object so that
+        # the None value is matched in the index
+        # https://github.com/pandas-dev/pandas/issues/55834
+        idx = Index(["a", "b", None], dtype="object")
+        result = idx.get_indexer([None, "x"])
+        expected = np.array([2, -1], dtype=np.intp)
+        tm.assert_numpy_array_equal(result, expected)
+
 
 class TestGetIndexerNonUnique:
     def test_get_indexer_non_unique_nas(self, nulls_fixture):