diff --git a/pandas/core/indexes/datetimelike.py b/pandas/core/indexes/datetimelike.py
index 7e6461f0fab5e..08588b8795668 100644
--- a/pandas/core/indexes/datetimelike.py
+++ b/pandas/core/indexes/datetimelike.py
@@ -637,6 +637,13 @@ def _can_fast_intersect(self, other: Self) -> bool:
             # Because freq is not None, we must then be monotonic decreasing
             return False
 
+        # for non-anchored frequencies, need to check that the two
+        #  indexes actually share a common point
+        # GH#44025
+        diff = other[0] - self[0]
+        if diff != Timedelta(0) and diff.total_seconds() % 86400 != 0:
+            return False
+
         # this along with matching freqs ensure that we "line up",
         #  so intersection will preserve freq
         # Note we are assuming away Ticks, as those go through _range_intersect
diff --git a/pandas/tests/indexes/datetimes/test_setops.py b/pandas/tests/indexes/datetimes/test_setops.py
index 7a68cb867c94e..230fd67bc9b82 100644
--- a/pandas/tests/indexes/datetimes/test_setops.py
+++ b/pandas/tests/indexes/datetimes/test_setops.py
@@ -7,6 +7,7 @@
 import numpy as np
 import pytest
 
+from pandas._libs.tslibs.timedeltas import Timedelta
 import pandas.util._test_decorators as td
 
 import pandas as pd
@@ -757,3 +758,16 @@ def test_intersection_non_nano_rangelike():
         freq="D",
     )
     tm.assert_index_equal(result, expected)
+
+
+def test_cday_intersection_empty():
+    # GH#44025
+    off = pd.offsets.CDay(1, normalize=False)
+    ts = Timestamp("2021-10-13 09:00")
+    ts2 = ts + Timedelta("1 hour")
+
+    dti1 = date_range(start=ts, periods=10, freq=off)
+    dti2 = date_range(start=ts2, periods=10, freq=off)
+    result = dti1.intersection(dti2)
+    expected = DatetimeIndex([], dtype="datetime64[ns]")
+    tm.assert_index_equal(result, expected)