Bidirectional real-time session sync between CLI and Web UI

Problem

Currently, Claude Code CLI and claude.ai web interface operate as isolated environments with only one-way session transfer:

• & prefix sends CLI context to web (one-way, creates new session)
• --teleport pulls web session into CLI (one-way)

There is no persistent bidirectional sync — users cannot work on the same session simultaneously from both interfaces, see each other's presence, or selectively load context between platforms.

Proposed Solution

Extend the existing teleportation mechanism into a persistent shared session:

Core Features

1. Shared Session ID — One conversation visible in both CLI and Web simultaneously
2. Real-time sync — Messages, tool results, and context updates propagate between platforms via WebSocket/SSE
3. Presence indicator — Show which clients are connected (CLI / Web / both)
4. Selective context loading — Choose which parts of the session to load (not the entire history, but specific segments on demand)
5. Skills/MCP parity — Skills and MCP servers work identically on both platforms

Proposed UX

# Start a shared session from CLI
claude --sync

# Connect to existing shared session from CLI
claude --sync <session-id>

# In web UI: "Open in Terminal" / "Sync with CLI" button

Architecture Concept

┌─────────────┐     Shared Session API      ┌──────────────┐
│  CLI Claude  │ ◄══════════════════════════►│  claude.ai   │
│  Code        │     (WebSocket/SSE)         │  Web UI      │
└──────┬──────┘                              └──────┬───────┘
       │                                            │
       ▼                                            ▼
┌───────────────────────────────────────────────────────────┐
│              Anthropic Session Registry                    │
│  - Unified session ID namespace                           │
│  - Sync state (messages, context, tools)                  │
│  - Presence tracking (who is connected)                   │
│  - Shared skills/MCP registry                             │
└───────────────────────────────────────────────────────────┘

Why This Matters

• Developers start work in CLI (heavy coding), then want to review/discuss in web UI (better for reading)
• Teams where one person uses CLI and another uses web need shared context
• Mobile continuation — start in CLI on desktop, continue in web on phone
• The existing & and --teleport prove session transfer is technically feasible — this extends it to be persistent and bidirectional

Related Issues

• #17682 — Cross-Environment Conversation History Sync
• #16593 — Auto-sync Sessions to Web
• #15881 — Seamless Session Sharing CLI-Desktop
• #15962 — Bi-directional Context Sync

This proposal unifies all of the above into a single coherent feature.

Additional Context

• Current workaround: third-party claude-sync tool that pushes/pulls ~/.claude/ to cloud storage
• MCP is not suitable for session sync (designed for tools, not conversations)
• The & prefix and --teleport already demonstrate that session context can cross platform boundaries