Remote-control session produces parallel divergent responses when resumed in VS Code

Bug Description

When a session initiated via a claude remote-control server is opened in VS Code while still active on the originating client (mobile browser), both clients begin generating independent response streams simultaneously. The two streams write interleaved entries into the same JSONL transcript file, producing two completely different reasoning chains for the same user message.

Steps to Reproduce

1. Start a remote control server with claude remote-control --name "Test"
2. Initiate a session from a mobile device and begin a multi-step conversation
3. While the session is mid-response, open VS Code and resume the same session from the chat history
4. Observe that from the attachment point onward, two independent assistant responses are generated in parallel

Observed Behavior

• The JSONL transcript contains interleaved entries from both sdk-cli and claude-vscode entrypoints
• Each stream generates its own independent reasoning, tool calls, and conclusions
• The two streams are unaware of each other — neither references or defers to the other's output
• The mobile client and VS Code client each display only their own stream, so the user sees two different conversations diverging from the same point
• One stream had different tool access than the other (e.g., one could run Bash, the other could not), leading to substantially different approaches

Expected Behavior

The session should behave like a shared conversation (similar to iMessage or a Gmail thread) — one unified conversation displayed across both devices, with a single response stream regardless of how many clients are viewing it.

Environment

• macOS
• Claude Code CLI + VS Code extension
• Session started via claude remote-control