[FEATURE] Bring Remote Control to the Desktop app's native Code sessions (parity with CLI)

Preflight

• [x] Searched existing issues — closest is #36795 (closed not planned / stale) and #46460 (Dispatch, a different feature)
• [x] Single feature request

Problem

Remote Control has been live and steadily maturing on the CLI (and VS Code) for months — claude remote-control, claude --remote-control, /remote-control. It lets me start a session on my machine and then pick it up and steer it from my phone at claude.ai/code or the Claude mobile app, with my real local filesystem, MCP servers, and project config intact.

The Desktop app still can't do this for its own native Code-tab sessions. Per the official "Choose the right approach" table, Remote Control runs on "Your machine (CLI or VS Code)" — Desktop is absent. So from the Desktop app, the only ways to reach a session from my phone are:

• Cloud / Remote sessions — these run in Anthropic's cloud, not on my machine with my local env. Different thing.
• Dispatch — a separate, heavier, more cluttered flow (message a task from mobile → it decides to spawn a session). It is not "take the live session I'm already running on my desktop and steer it from my phone."

What I want already exists on the CLI: open / continue a Desktop app session from my phone, exactly like claude remote-control. The Desktop app wraps the same CLI — there's no reason its GUI sessions can't register as Remote Control sessions and show up at claude.ai/code and in the mobile Code tab.

This was already filed as #36795 ("Desktop app sessions should be accessible via remote control") and closed as not planned / stale (v2.1.80, April). The gap is still here months later while CLI Remote Control has only gotten more capable, so I'm re-raising it. It's also distinct from #46460 (Dispatch on Desktop).

Proposed

• A per-session "Enable Remote Control" control on native Desktop Code sessions (and honoring remoteControlAtStartup / the "Enable remote control by default" setting), so a Desktop session registers just like claude --remote-control.
• Desktop sessions then appear in the session list at claude.ai/code and the mobile Code tab (computer icon + green dot), fully steerable from the phone.

Expected result

Full parity: a session I start in the Desktop app is one I can pick up and drive from my phone — without falling back to Dispatch or cloud sessions.

Environment

Claude Code Desktop app, Windows 11 (ARM64 / Snapdragon).