gh-137597: Clarify flattening and bugfix for itertools.tee

Doc/library/itertools.rst

@@ -731,6 +731,17 @@ loops that truncate the stream.
    allows nested :func:`tee` calls to share the same underlying data
    chain and to have a single update step rather than a chain of calls.
 
+   .. note::
+
+    :func:`tee` automatically "flattens" existing tee objects,
+    sharing the same underlying buffer instead of nesting them, to avoid
+    performance degradation.
+
+   .. versionchanged:: 3.13
+      Fixed a bug where re-teeing a tee iterator did not always return independent iterators.
+      Previously, the first iterator in the result could be the same object as the
+      input, causing inconsistent behavior.
+
    The flattening property makes tee iterators efficiently peekable:
 
    .. testcode::