Realtime: send session update before tool response

Author: rm-openai

src/agents/realtime/session.py

@@ -374,19 +374,20 @@ async def _handle_tool_call(self, event: RealtimeModelToolCallEvent) -> None:
                 )
             )
 
-            # Send tool output to complete the handoff
+            # First, send the session update so the model receives the new instructions
+            await self._model.send_event(
+                RealtimeModelSendSessionUpdate(session_settings=updated_settings)
+            )
+
+            # Then send tool output to complete the handoff (this triggers a new response)
+            transfer_message = handoff.get_transfer_message(result)
             await self._model.send_event(
                 RealtimeModelSendToolOutput(
                     tool_call=event,
-                    output=f"Handed off to {self._current_agent.name}",
+                    output=transfer_message,
                     start_response=True,
                 )
             )
-
-            # Send session update to model
-            await self._model.send_event(
-                RealtimeModelSendSessionUpdate(session_settings=updated_settings)
-            )
         else:
             raise ModelBehaviorError(f"Tool {event.name} not found")