diff --git a/pandas/conftest.py b/pandas/conftest.py
index e2db9260ac37d..7ad322d050c0f 100644
--- a/pandas/conftest.py
+++ b/pandas/conftest.py
@@ -667,7 +667,8 @@ def _create_mi_with_dt64tz_level():
 
 
 indices_dict = {
-    "string": Index([f"pandas_{i}" for i in range(10)]),
+    "object": Index([f"pandas_{i}" for i in range(10)], dtype=object),
+    "string": Index([f"pandas_{i}" for i in range(10)], dtype="str"),
     "datetime": date_range("2020-01-01", periods=10),
     "datetime-tz": date_range("2020-01-01", periods=10, tz="US/Pacific"),
     "period": period_range("2020-01-01", periods=10, freq="D"),
diff --git a/pandas/tests/indexes/test_any_index.py b/pandas/tests/indexes/test_any_index.py
index e1ed96195e0a7..a4c18732ef258 100644
--- a/pandas/tests/indexes/test_any_index.py
+++ b/pandas/tests/indexes/test_any_index.py
@@ -40,7 +40,7 @@ def test_map_identity_mapping(index, request):
     # GH#12766
 
     result = index.map(lambda x: x)
-    if index.dtype == object and result.dtype == bool:
+    if index.dtype == object and (result.dtype == bool or result.dtype == "string"):
         assert (index == result).all()
         # TODO: could work that into the 'exact="equiv"'?
         return  # FIXME: doesn't belong in this file anymore!
diff --git a/pandas/tests/indexes/test_old_base.py b/pandas/tests/indexes/test_old_base.py
index 0199e21bfc980..65feb07e05d9f 100644
--- a/pandas/tests/indexes/test_old_base.py
+++ b/pandas/tests/indexes/test_old_base.py
@@ -256,7 +256,7 @@ def test_ensure_copied_data(self, index):
                 "RangeIndex cannot be initialized from data, "
                 "MultiIndex and CategoricalIndex are tested separately"
             )
-        elif index.dtype == object and index.inferred_type == "boolean":
+        elif index.dtype == object and index.inferred_type in ["boolean", "string"]:
             init_kwargs["dtype"] = index.dtype
 
         index_type = type(index)
diff --git a/pandas/tests/indexes/test_setops.py b/pandas/tests/indexes/test_setops.py
index e5dc47be20677..5f934ca3e6e83 100644
--- a/pandas/tests/indexes/test_setops.py
+++ b/pandas/tests/indexes/test_setops.py
@@ -299,7 +299,13 @@ def test_difference_base(self, sort, index):
                 first.difference([1, 2, 3], sort)
 
     @pytest.mark.filterwarnings(r"ignore:PeriodDtype\[B\] is deprecated:FutureWarning")
-    def test_symmetric_difference(self, index):
+    def test_symmetric_difference(self, index, using_infer_string, request):
+        if (
+            using_infer_string
+            and index.dtype == "object"
+            and index.inferred_type == "string"
+        ):
+            request.applymarker(pytest.mark.xfail(reason="TODO: infer_string"))
         if isinstance(index, CategoricalIndex):
             pytest.skip(f"Not relevant for {type(index).__name__}")
         if len(index) < 2:
diff --git a/pandas/tests/test_algos.py b/pandas/tests/test_algos.py
index dac74a0e32a42..81e7d3774b613 100644
--- a/pandas/tests/test_algos.py
+++ b/pandas/tests/test_algos.py
@@ -65,6 +65,7 @@ def test_factorize_complex(self):
         expected_uniques = np.array([(1 + 0j), (2 + 0j), (2 + 1j)], dtype=complex)
         tm.assert_numpy_array_equal(uniques, expected_uniques)
 
+    @pytest.mark.xfail(using_string_dtype(), reason="TODO(infer_string)", strict=False)
     def test_factorize(self, index_or_series_obj, sort):
         obj = index_or_series_obj
         result_codes, result_uniques = obj.factorize(sort=sort)