diff --git a/src/mcp/client/streamable_http.py b/src/mcp/client/streamable_http.py
index aa3e50e07..8416874e2 100644
--- a/src/mcp/client/streamable_http.py
+++ b/src/mcp/client/streamable_http.py
@@ -241,15 +241,17 @@ async def _handle_resumption_request(self, ctx: RequestContext) -> None:
             logger.debug("Resumption GET SSE connection established")
 
             async for sse in event_source.aiter_sse():  # pragma: no branch
-                is_complete = await self._handle_sse_event(
+                await self._handle_sse_event(
                     sse,
                     ctx.read_stream_writer,
                     original_request_id,
                     ctx.metadata.on_resumption_token_update if ctx.metadata else None,
                 )
-                if is_complete:
-                    await event_source.response.aclose()
-                    break
+                # Drain to EOF rather than breaking on completion (see
+                # _handle_sse_response): breaking leaves the enclosing
+                # `async with aconnect_sse(...)` to close an un-drained stream,
+                # which can stall the next request that reuses the connection.
+                # Cancellation still tears the stream down promptly.
 
     async def _handle_post_request(self, ctx: RequestContext) -> None:
         """Handle a POST request with response processing."""
@@ -340,6 +342,8 @@ async def _handle_sse_response(
         assert isinstance(ctx.session_message.message, JSONRPCRequest)
         original_request_id = ctx.session_message.message.id
 
+        response_complete = False
+
         try:
             event_source = EventSource(response)
             async for sse in event_source.aiter_sse():  # pragma: no branch
@@ -358,16 +362,20 @@ async def _handle_sse_response(
                     resumption_callback=(ctx.metadata.on_resumption_token_update if ctx.metadata else None),
                     is_initialization=is_initialization,
                 )
-                # If the SSE event indicates completion, like returning response/error
-                # break the loop
+                # Once the response/error arrives the caller is unblocked, but we keep
+                # iterating so the SSE body is drained to EOF before the connection is
+                # released. Closing the response early (await response.aclose()) leaves
+                # the keepalive connection un-drained, which can stall the next request
+                # that reuses it. Cancellation still tears down promptly: CancelledError
+                # propagates out of aiter_sse() and the caller's
+                # `async with client.stream(...)` closes the response.
                 if is_complete:
-                    await response.aclose()
-                    return  # Normal completion, no reconnect needed
+                    response_complete = True
         except Exception:
             logger.debug("SSE stream ended", exc_info=True)  # pragma: no cover
 
-        # Stream ended without response - reconnect if we received an event with ID
-        if last_event_id is not None:  # pragma: no branch
+        # Stream ended without a response - reconnect if we received an event with ID
+        if not response_complete and last_event_id is not None:
             logger.info("SSE stream disconnected, reconnecting...")
             await self._handle_reconnection(ctx, last_event_id, retry_interval_ms)
 
@@ -404,6 +412,7 @@ async def _handle_reconnection(
                 # Track for potential further reconnection
                 reconnect_last_event_id: str = last_event_id
                 reconnect_retry_ms = retry_interval_ms
+                response_complete = False
 
                 async for sse in event_source.aiter_sse():
                     if sse.id:  # pragma: no branch
@@ -417,9 +426,12 @@ async def _handle_reconnection(
                         original_request_id,
                         ctx.metadata.on_resumption_token_update if ctx.metadata else None,
                     )
+                    # Drain to EOF instead of closing early (see _handle_sse_response).
                     if is_complete:
-                        await event_source.response.aclose()
-                        return
+                        response_complete = True
+
+                if response_complete:
+                    return
 
                 # Stream ended again without response - reconnect again (reset attempt counter)
                 logger.info("SSE stream disconnected, reconnecting...")