[FEATURE] Show Remote Control status in the agent view / session lists

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

When running multiple Claude Code sessions, the agent view and session lists give no indication of which sessions have Remote Control enabled, and the Remote Control session name is not shown anywhere locally.

Today the only places Remote Control status is visible are:

• Inside an individual session (the /rc active footer indicator, or the /remote-control panel)
• The claude.ai/code web session list (computer icon with a green status dot)

Neither the agent view, the desktop sidebar, nor the /resume picker shows it. The name given via /remote-control <name> (or --remote-control [name]) exists only on the claude.ai side — locally the session keeps its auto-derived display name, so there is no way to match "the session I named Reviews on the web" back to a local session.

Concrete scenario where this bites: I had a session with Remote Control enabled (named "Reviews") so I could act on PR reviews from my phone. After a machine restart the process died and the web session shows "disconnected". I wanted to resume that exact session — but in the agent view and the --resume picker there is nothing to identify it: no RC indicator, no RC session name, dozens of similar-looking sessions. I ended up grepping ~/.claude/history.jsonl for the /remote-control command to recover the session ID.

Proposed Solution

Show a Remote Control indicator per session in the agent view, desktop sidebar, and /resume picker — for example the same computer icon + green/grey status dot already used on claude.ai/code, or a short RC badge next to the session name (green when connected, grey when the conversation has an RC session recorded but is currently offline, absent when never enabled).

Additionally, surface the Remote Control session name locally (e.g. Reviews · RC next to the display name), and make the status filterable wherever session filtering already exists (the desktop sidebar can already filter by status/project/environment — Remote Control would fit naturally alongside those).

Notably, the data already exists locally: the per-session state files in ~/.claude/sessions/*.json carry a bridgeSessionId field for RC-connected sessions alongside name/nameSource — it just isn't rendered in any list view.

Alternative Solutions

• Using claude.ai/code as a separate "which sessions are remote-controlled" dashboard — works only while sessions are alive, and means leaving the terminal and mentally joining two lists.
• Manually naming sessions (--name / /rename) with an RC prefix when enabling Remote Control — fragile and easy to forget.
• Grepping ~/.claude/history.jsonl for /remote-control commands to recover the session ID — works, but is not a discoverable or supported workflow.
• Issue #31840 proposes exposing Remote Control state via a state file / CLI flag for external tooling; that would enable scripting a workaround but doesn't address the built-in views.

Priority

Medium - Would be very helpful

Feature Category

Interactive mode (TUI)

Use Case Example

1. I have 4–6 Claude Code sessions running across different repos; a couple have Remote Control enabled so I can continue them from my phone (e.g. a PR-review session running /loop).
2. My machine restarts; every process dies and the RC sessions show "disconnected" on claude.ai.
3. I want to resume the specific session that had Remote Control enabled — not start a new one — so my review workflow keeps its web session and history.
4. Today neither the agent view nor the --resume picker shows which session that was; with this feature, the RC badge/name would identify it at a glance and I could resume it, re-enable RC, and restart the loop in seconds.

Additional Context

• Related: #31840 ([FEATURE] Expose Remote Control connection status via state file or CLI flag) — complementary; that issue covers machine-readable state for external tooling, this one covers the built-in session views.
• Related but distinct (already filed, not duplicated here): #60790 / #36401 (RC session shows disconnected after the owning process dies), #61735 (ScheduleWakeup/loop not persistent across restarts), #26249 / #39414 / #25841 (resume picker doesn't show/index session names). This request is only about visibility of RC status + name in the session lists.
• The claude.ai/code web list already renders this state (computer icon + green dot per the Remote Control docs), and ~/.claude/sessions/*.json already records bridgeSessionId locally — so this is a rendering gap, not a data gap.