Cross-client session syncing between VS Code extension and Desktop app

Summary

Sessions started in the VS Code extension don't appear in the Claude Desktop app's Code tab, and vice versa. There's no way to start a session in one client and continue it in the other, or to see a unified list of all sessions regardless of where they were started.

Use case

I work across both the VS Code extension (for inline IDE integration) and the Claude Desktop app (for standalone work). I'd like to:

• See all my Claude Code sessions in a single list, regardless of which client started them
• Start a session in VS Code and pick up where I left off in the Desktop app (and vice versa)

Current behavior

• VS Code extension and Desktop app are fully siloed — each only shows sessions it started
• Session transcripts all land in ~/.claude/projects/ on disk, but neither client surfaces the other's sessions
• The CLI claude --resume can pick up any session, but the GUI clients cannot

Proposed behavior

• The Code tab in the Desktop app (and the VS Code extension's session list) should read from the shared ~/.claude/projects/ transcript store and display all sessions
• Resuming a session started in a different client should work seamlessly