Updated condition logic

Author: arthurlw

pandas/core/indexes/datetimelike.py

@@ -626,8 +626,7 @@ def _fast_intersect(self, other, sort):
         return result
 
     def _can_fast_intersect(self, other: Self) -> bool:
-        # Note: we only get here with len(self) > 0 and len(other) > 0
-        if self.freq is None or self.freq == "C":
+        if self.freq is None:
             return False
 
         elif other.freq != self.freq:
@@ -637,6 +636,13 @@ 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