Support cmux as a terminal backend for agent teams

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

Claude Code agent teams currently only support tmux and iTerm2 as terminal backends for spawning teammate panes. Users of cmux — a native macOS terminal built on Ghostty's rendering engine and designed specifically for AI coding agent workflows — are forced to either:

• Fall back to in-process mode, losing all pane visibility into teammate activity
• Nest tmux inside cmux, adding unnecessary complexity

There is no teammateMode option to use cmux's pane management, despite cmux exposing a CLI and socket API purpose-built for this kind of automation.

Proposed Solution

Add cmux as a supported teammateMode option that:

1. Detects cmux availability via its CLI or socket
2. Creates split panes for each teammate using the cmux socket/CLI API
3. Routes teammate I/O through cmux panes
4. Supports the same lifecycle operations (create, monitor, shutdown) as the tmux backend

Configuration would look like:

{
  "teammateMode": "cmux"
}

The auto mode should also detect cmux and prefer it when available.

Alternative Solutions

• Nesting tmux inside cmux — works but adds complexity and loses cmux's native features (notification rings, git context in tabs, embedded browser)
• Running separate Claude Code sessions in different cmux workspaces — works but loses unified team coordination
• Using in-process mode — functional but provides no visibility into teammate activity

Priority

Medium - Would be very helpful

Feature Category

Interactive mode (TUI)

Use Case Example

Example scenario:

1. I'm working in cmux as my primary terminal, using its vertical tabs and split panes for my workflow
2. I spin up a Claude Code agent team to parallelize a task across 3 teammates
3. Currently, the teammates either spawn invisibly (in-process) or I have to run tmux inside cmux
4. With this feature, each teammate would appear in a native cmux split pane with notification rings showing when they need attention
5. This would provide a better UX than tmux panes since cmux shows git branch, working directory, and port info per-pane

Additional Context

• cmux website: https://www.cmux.dev/
• cmux GitHub: https://github.com/manaflow-ai/cmux
• cmux socket API docs: https://www.cmux.dev/docs/api
• cmux already documents integration with Claude Code hooks for notifications, indicating the developers are actively targeting this use case