[FEATURE] Extend Remote Control to the VSCode extension

Preflight Checklist

• [x] I have searched existing requests and this feature hasn't been requested yet
• [x] This is a single feature request (not multiple features)

Problem Statement

Summary

Claude Code's Remote Control feature (v2.1.121) lets terminal CLI sessions be mirrored to claude.ai/code and the mobile app, but the VSCode extension can't participate. Please extend Remote Control to the VSCode extension so IDE sessions can be continued from a phone or browser.

Why this matters

The VSCode extension and the terminal CLI have effectively the same capabilities (file edits, MCP, skills, plugins), but they're operationally split for mobile users:

VSCode users get a richer desktop UX (inline diffs in editor tabs, file tree integration, IDE-aware context, native multi-file editing)
CLI users get phone continuity via Remote Control
Today, choosing one means giving up the other. Many of us would prefer to use the VSCode extension at our desks AND pick up the same session on a phone when we step away — exactly the workflow Remote Control is designed for.

Proposed Solution

A "Start Remote Control" command in the VSCode command palette (and/or /remote-control in the extension's chat input) that produces a session URL + QR code, same as the CLI.
The active session appears in the Code tab on mobile and at claude.ai/code with a green online indicator.
Bidirectional sync: messages typed in the IDE appear on the phone and vice versa, with tool results streaming to both.
Tool execution still happens locally on the user's Mac (same security model as the CLI version).

Alternative Solutions

Running claude --remote-control "name" in a terminal — works, but means losing the VSCode UI for that project.
Pasting context into a fresh claude.ai/code session — works for short tasks, but breaks down for anything stateful (loaded files, MCP, in-progress work).

Priority

High - Significant impact on productivity

Feature Category

Interactive mode (TUI)

Use Case Example

Long-running tasks — kick off a refactor or test run in VSCode, leave the office, monitor and steer it from the phone on the train.
Cross-device continuity — a session with full project context (loaded files, MCP state, conversation history) carried forward without copy-pasting a "here's what we discussed" recap into a fresh CLI session.
Single source of truth — one conversation history per project instead of parallel VSCode and CLI threads that drift apart.
Off-hours triage — review a build failure or PR comment on the phone using the same agent that already has project context loaded, without spinning up a new terminal session at home.

Additional Context

Environment

Claude Code v2.1.121
macOS Darwin 25.3.0
VSCode extension + terminal CLI both installed