[BUG] teammateMode: "tmux" — split panes open but teammates are disconnected from messaging system

Bug Description

When using --teammate-mode tmux with tmux -CC in iTerm2, agent team split panes do open and separate Claude Code sessions do launch in each pane, but the teammate sessions are completely disconnected from the lead's messaging and task coordination system.

This is distinct from the silent fallback issues reported in #23572, #23815, #24292, and #24301 where panes don't open at all. In this case, the panes open successfully but the teammate processes inside them never receive instructions.

Reproduction Steps

1. macOS, iTerm2, tmux 3.6a installed, it2 CLI installed, iTerm2 Python API enabled
2. Start tmux control mode: tmux -CC
3. Launch Claude Code: claude --teammate-mode tmux
4. Ask Claude to create an agent team (e.g., "Create an agent team with 2 teammates")
5. Observe: Split panes open, each with a Claude Code session showing the startup banner and idle > prompt

Expected Behavior

Teammate panes should show real-time output of the teammates working — tool calls, thinking, streaming output — as the lead assigns them tasks.

Actual Behavior

• Split panes open with Claude Code sessions (correct)
• Teammate panes show idle > prompt with zero output — no scrollback, no activity
• Lead pane reports teammates are "mid-work" and "active in tmux panes"
• Lead's SendMessage to teammates is not received by the pane sessions
• Lead uses tmux capture-pane to check and confirms panes are "sitting idle at their prompts"
• Lead attempts to send instructions directly via SendMessage(@teammate) — messages are not picked up
• Lead ultimately gives up and does the work directly

Key Observation

The lead ran tmux capture-pane -t %12 -p -S -50 and tmux capture-pane -t %13 -p -S -50 which showed the Claude Code startup banners in the panes, confirming sessions launched. But those sessions have no connection to the team's messaging/task system.

Environment

• Claude Code: v2.1.32
• OS: macOS Darwin 25.2.0 (arm64)
• Terminal: iTerm2
• tmux: 3.6a
• it2 CLI: installed at /Users/jaime/miniconda3/bin/it2
• iTerm2 Python API: enabled
• Launch method: tmux -CC → claude --teammate-mode tmux

Related Issues

• #23572 — Silent fallback to in-process (different: panes don't open at all)
• #23815 — iTerm2 split-pane spawns in-process (different: panes don't open at all)
• #24292 — Duplicate of #23815
• #24301 — Silent fallback (different: panes don't open at all)
• #23615 — Panes split but send-keys corruption (related but different symptom)

Summary

The split pane creation works. The Claude Code session launch in panes works. But the routing of teammate processes to those pane sessions is broken — the panes are orphaned instances with no connection to the team coordination layer.