Sync local CLI session history to the cloud for cross-device resume (parity with Codex)

Problem

Local Claude Code CLI sessions (~/.claude/projects/**/*.jsonl) are stored per-machine and are not synced anywhere. When I work across 2 desktop PCs + a phone, I can't see or resume a session I started on one machine from another device.

The existing cross-device options don't cover this:

• Remote Control requires the host PC to be running and reachable. If the machine is asleep or off, the session is gone.
• Claude Code on the web / cloud sessions are great, but they run in a cloud VM and can't touch my local files (local dev servers, PM2 processes, a synced project folder on my own disk).

So there is no way to "run locally, then pick the same history up on another device."

Request

Add an opt-in setting to sync local CLI session history to the cloud, tied to my account, so any signed-in device can list and resume those sessions — even asynchronously, without the original host machine being online.

Why it matters

OpenAI Codex already does this: run it locally, and the session history shows up on every device signed into the same account. It's the one cross-device gap that stops me from using Claude Code's local CLI as my hub across my machines. An opt-in toggle (so it respects privacy for those who don't want uploads) would fully close the gap.

Notes

• Opt-in / off by default is fine — I understand many users won't want local transcripts uploaded.
• Even a "list + read-only view from other devices" would be a big improvement; full resume would be ideal.