[FEATURE] Add `/fork` and `/branch` (conversation branching) support to Remote Control web/mobile clients

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

The CLI supports /branch (clone the conversation, switch into the new copy, leave the original untouched) and /fork <directive> (spawn a background subagent that inherits the conversation and works independently). This has already been requested for other surfaces that lack it: the VS Code extension (#69272), the JetBrains plugin (#93301), and the Cursor extension (#93379).

Remote Control has the same gap, for both commands:

• Per the Remote Control docs, under Limitations > "Some commands are local-only," neither /fork nor /branch is in the list of commands supported from mobile/web during a Remote Control session (only /compact, /clear, /context, /usage, /exit, /usage-credits, /recap, /reload-plugins, /model, /effort, /fast, /color, /rename, /mcp, /config, /autocompact, and /advisor are listed as working remotely).
• Attempting /fork from claude.ai/code or the Claude mobile app while connected via Remote Control fails at runtime with a generic error, "Couldn't fork this session," instead of being disabled or explained.
• /branch doesn't even produce an error — it's simply not recognized as a command at all in the web view, with no feedback of any kind.

Proposed Solution

• Add /fork and /branch support to Remote Control's web and mobile clients, matching their CLI behavior.
• At minimum, if full support isn't feasible soon, both commands should fail with a clear message stating they're local-only (consistent with how /resume and /plugin are documented), rather than /fork's generic runtime error or /branch's silent non-recognition.

Alternative Solutions

For claude remote-control (server mode), there is no workaround at all: the local terminal only displays connection status and logs, with no input prompt to type /fork or /branch into locally. The only way to reach either command is to stop server mode and start an interactive session instead (claude --remote-control), which does have a local prompt — but that changes the whole session model and isn't a real substitute for server mode's multi-session design.

Priority

Medium - Would be very helpful

Feature Category

Other

Use Case Example

1. I'm steering a session on my home machine via claude.ai/code from my phone, with claude remote-control running in server mode.
2. Claude proposes an approach mid-session and I want to try a different direction without losing the current one.
3. On the CLI I'd /branch to get a separate session I can go back and forth between, or /fork <directive> to delegate a variation as a background task.
4. From Remote Control's web view, I can't do either: /fork fails with "Couldn't fork this session," and /branch isn't recognized as a command at all. There's also no local terminal prompt to fall back to, since server mode only shows logs.

Additional Context

• Same underlying request as #69272 (VS Code), #93301 (JetBrains), #93379 (Cursor) — this covers the Remote Control web/mobile surface specifically.
• Environment: claude remote-control (server mode) on a self-hosted machine, attempting /fork and /branch from claude.ai/code (web view).