Describe the Bug

When running the MCP server behind a proxy and exposing it at a subpath such as https://example.com/mcp/sse, the EventSource connection is established successfully. However, subsequent redirection to the /messages/* endpoint does not preserve the base path prefix.

Instead of using the correct path https://example.com/mcp/messages/*, the SDK incorrectly attempts to connect to https://example.com/messages/*, stripping the /mcp prefix.

Relevant code:

Steps to Reproduce

1. Deploy the MCP server behind a reverse proxy with a path prefix (e.g., /mcp).
2. Access the SSE endpoint at https://example.com/mcp/sse.
3. Observe the redirected connection target when the server pushes updates.

Expected Behavior

The SDK should preserve the original base path prefix (/mcp) and construct the /messages/* URLs relative to it, resulting in a correct path like https://example.com/mcp/messages/*.