Sync sessions across Claude Code CLI, Desktop app, and web

Summary

Claude Code CLI sessions and the Claude Desktop app each maintain their own independent session history (per https://code.claude.com/docs/en/sessions.md). There's no way to view or continue a CLI conversation from the Desktop app, or vice versa. The only handoff today is /export + manual paste, which is lossy and one-shot.

Use case

I run long Claude Code CLI sessions while working in VR (Quest 3 game dev) on my desktop. I'd like to glance at the conversation on my phone or laptop via the Desktop app — or hand off the active session to continue there without losing context.

Suggested directions (any of these would help)

1. Account-scoped session sync — same conversation visible in CLI, Desktop, web
2. Read-only live view of an active CLI session from the Desktop app
3. Resume a CLI session in the Desktop app (and vice versa)

Workarounds today

/export -> paste into Desktop. Lossy, one-shot, no live view