client's `read_stream_writer` open after SSE disconnection hanging `.receive()`

Initial Checks

• I confirm that I'm using the latest version of MCP Python SDK
• I confirm that I searched for my issue in https://github.com/modelcontextprotocol/python-sdk/issues before opening this issue

Description

This issue MAY be related to the following

• https://github.com/modelcontextprotocol/python-sdk/issues/1805
• https://github.com/modelcontextprotocol/python-sdk/issues/1764
• https://github.com/modelcontextprotocol/python-sdk/issues/262

Experiencing deadlocks on streamable_http transport. In order to reproduce the issue the following can be run.

import asyncio
import logging
import threading
import time

from fastmcp import Client, FastMCP
from fastmcp.client.transports import StreamableHttpTransport

logging.basicConfig(
    level=logging.DEBUG,
    format="%(asctime)s - %(name)s - %(levelname)s - %(message)s",
    handlers=[logging.StreamHandler()],
)

logger = logging.getLogger(__name__)

HOST = "127.0.0.1"
PORT = 8765
SERVER_URL = f"http://{HOST}:{PORT}/mcp"

mcp = FastMCP(name="timeout issue")

SSE_TIMEOUT = 0.1
SLEEP = 60


@mcp.tool
def blocking_call() -> str:
    time.sleep(SLEEP)
    return "42"


def run_server():
    mcp.run(transport="streamable-http", host=HOST, port=PORT)


async def test_blocking_sse():
    transport = StreamableHttpTransport(SERVER_URL, sse_read_timeout=SSE_TIMEOUT)
    async with Client(transport) as client:
        tools = await client.list_tools()
        print(f"available tools: {[t.name for t in tools]}")
        result = await client.call_tool("blocking_call", {})
        print(f"blocking result: {result}")


if __name__ == "__main__":
    server_thread = threading.Thread(target=run_server, daemon=True)
    server_thread.start()

    time.sleep(2)

    try:
        asyncio.run(test_blocking_sse())
    finally:
        logger.info(f"{time.strftime('%Y-%m-%d %H:%M:%S')} [CLIENT] Shutting down...")

Execution context := locally built fastmcp at this commit 790ea92

Observed Behavior: the client hangs forever after the SSE read timeout fires, looking at the logs:

• tool request is sent, server returns HTTP 200 with [Content-Type: text/event-stream]
• server starts executing the tool (blocking time.sleep(60))
• client's SSE read timeout fires (after ~5 seconds with default httpx timeout)
• client closes the HTTP connection
• client hangs indefinitely - call_tool() never returns
• the tool eventually completes on the server side, but when the server tries to send the response back, it gets BrokenResourceError because the HTTP connection was already closed by the client.

I know noting about nothing, so I can imagine the above example is just an issue on my side, maybe:

• sse timeout should always be greater or equal to the expected timeout of tool calls (now that sse_read_timeout is deprecated it's httpx.Timeout counterpart should be >= tool timeout)
• long running tool calls MUST send progress updates

but one thing is for sure and that is, the above configuration hangs indefenitely because the session layer never gets to know that the transport layer is dead after the SSE stream is closed

Within my ignorance of many aspects of the implementation, the missing else branch seems to be the root cause.

https://github.com/modelcontextprotocol/python-sdk/blob/a9cc822a1051b1bd2b6b9b57e9e4136406983b61/src/mcp/client/streamable_http.py#L433-L435

and the fix looks something like

if last_event_id is not None:  # pragma: no branch 
    ...
else:
    error_response = JSONRPCError(...)
    await ctx.read_stream_writer.send(SessionMessage(JSONRPCMessage(root=error_response)))

consequently raising McpError

Example Code

Python & MCP Python SDK

Execution context for ease of implementation 
- `mcp==1.24.0`
- example codebase run against locally built `fastmcp` at [790ea92](https://github.com/jlowin/fastmcp/tree/790ea92eb59256da68c83097321ebde8f8819bcf) --> 1.24.0
- Python: `python-3.12.7-macos-aarch64-none/bin/python3.12`

从 src/mcp/client/streamable_http.py 的第 433-435 行附近开始，使用 StreamableHttpTransport 的 Python 示例复现挂起的调用。跟踪 SSE 读取超时如何关闭连接，以及 session 如何消费 read_stream_writer 消息。完成的标准是：超时的客户端调用能够返回或抛出异常，而不是无限期挂起。