diff --git a/Lib/asyncio/sslproto.py b/Lib/asyncio/sslproto.py
index 541e252..31eab51 100644
--- a/Lib/asyncio/sslproto.py
+++ b/Lib/asyncio/sslproto.py
@@ -572,8 +572,12 @@
             # wait until protocol.connection_made() has been called
             self._waiter._set_result_unless_cancelled(None)
         self._session_established = True
-        # In case transport.write() was already called
-        self._process_write_backlog()
+        # In case transport.write() was already called. Don't call
+        # immediatly _process_write_backlog(), but schedule it:
+        # _on_handshake_complete() can be called indirectly from
+        # _process_write_backlog(), and _process_write_backlog() is not
+        # reentrant.
+        self._loop.call(self._process_write_backlog)
 
     def _process_write_backlog(self):
         # Try to make progress on the write backlog.