Streamable HTTP transport drops requests immediately after `initialize`

[AydarDD]

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

Solution: #1674

Title: Streamable HTTP transport drops requests immediately after initialize

Summary
When using a server over Streamable HTTP, the first session.list_tools() (and sometimes a few follow-up requests) can intermittently return an empty tool list right after session.initialize() succeeds. This happens even though the server has already provided tool metadata during initialization.

Steps to Reproduce

1. Start examples/servers/simple-streamablehttp.
2. Run a client that calls session.initialize() and immediately follows with session.list_tools().
3. Repeat quickly; roughly 1 in 5 iterations returns an empty list.

Expected Behavior
Once initialize completes, the client transport should be ready to send subsequent JSON-RPC requests and receive their responses reliably.

Actual Behavior
There is a race inside streamable_http.streamablehttp_client: the post_writer task is started with tg.start_soon, so the caller can enqueue requests before the writer task has finished subscribing to the in-memory stream. Because the stream buffer size is 0, those requests can be dropped, leading to empty responses or timeouts.

Proposed Fix
Start post_writer with tg.start(...), mirroring the SSE transport. This blocks until the writer task signals readiness via TaskStatus.started, guaranteeing that the zero-buffer stream is ready before yielding to the caller. Two new stress tests (test_streamablehttp_no_race_condition_on_consecutive_requests and test_streamablehttp_rapid_request_sequence) cover the regression.

Impact
Streamable HTTP transports become much more reliable in real deployments that issue back-to-back requests (initialize → list_tools, quick polling, etc.), preventing confusing empty tool listings and retry storms.

Example Code

# reproduce_streamablehttp_race.py
import anyio

from mcp.client.session import ClientSession
from mcp.client.streamable_http import streamablehttp_client


SERVER_URL = "http://127.0.0.1:8000/mcp"
ITERATIONS = 100


async def main() -> None:
    failures = 0

    for i in range(ITERATIONS):
        async with streamablehttp_client(SERVER_URL) as (read_stream, write_stream, _):
            async with ClientSession(read_stream, write_stream) as session:
                result = await session.initialize()
                tools = await session.list_tools()
                if not tools.tools:
                    failures += 1
                    print(f"[{i}] list_tools returned EMPTY! session.initialize -> list_tools race hit.")
                else:
                    print(f"[{i}] ok: {len(tools.tools)} tools")

    if failures:
        raise SystemExit(f"Race reproduced: {failures}/{ITERATIONS} iterations failed")

    print("No failures observed (try increasing ITERATIONS or lower server CPU).")


if __name__ == "__main__":
    anyio.run(main)

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