[FEATURE] Desktop: option to sync archive state between a Remote Control session and its host machine

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

I use Claude Code Desktop on two Windows machines with the same account: a laptop, and a desktop PC (the host) where sessions actually run. With "Connect new sessions to Remote Control" turned on on the host, every host session also appears in the laptop's Code sidebar, which is great for working from the laptop.

But archive state is per device. When I archive a finished session from the laptop, it only disappears from the laptop. The host still lists it as active, so I have to archive every session a second time on the host (through Remote Desktop). The two lists drift apart and the host's sidebar keeps growing with sessions I already cleaned up.

Proposed Solution

Add a setting in Settings → Claude Code, next to "Connect new sessions to Remote Control", for example "Sync archive state with Remote Control devices" (on/off). When it is on:

• Archiving a Remote Control session from any device (another Desktop app, the web, mobile) also archives the local session on the host machine.
• Archiving a session on the host also archives it on every device that shows it.
• Unarchiving works the same way in both directions.

To keep it safe, the host's Archived filter could show "Archived from [device name]" so a session archived by mistake is easy to find and restore.

Alternative Solutions

• Archiving twice by hand, on the laptop and then on the host. This is what I do today, and it's tedious with 30+ sessions.
• Asking Claude in a host session to archive a list of sessions. It works, but it's still a second pass and needs one approval per session.
• "Auto-archive after PR merge or close" only covers sessions that have a PR.
• Turning off "Connect new sessions to Remote Control" removes the duplicates, but I lose the ability to steer host sessions from the laptop.

Priority

Medium - Would be very helpful

Feature Category

Configuration and settings

Use Case Example

1. On my host PC, Claude Desktop runs about 30 Code sessions with Remote Control on.
2. During the day I work in those sessions from my laptop.
3. When a task is done, I archive its session from the laptop.
4. When I next open the host (via Remote Desktop), all those sessions are still in its active list, and I have to find and archive each one again.

With the option on, step 4 would no longer be needed.

Additional Context

• Related: #97981 (host sidebar keeps listing sessions archived on another machine, macOS, open bug) and #65838 (archive state is per client, iOS ↔ macOS, closed as not planned after inactivity). This request asks for an explicit, user-controlled option, so the behavior is predictable even if the default stays per device.
• Environment: Claude Desktop on Windows 11 on both machines, same account.