Feature: Access Claude Code desktop projects remotely from the mobile app

Summary

When a Claude Code project is connected to a remote resource (e.g. a cloud-hosted repo, a remote server, a Supabase instance, a Cloudflare Worker) and doesn't depend on local desktop files or processes, there's no reason it should be desktop-only. The mobile app should be able to open and continue working in those projects directly.

Use case

A significant portion of real-world Claude Code projects are essentially remote-first:

• Web apps hosted on Cloudflare Pages / Vercel / Netlify
• Backend services with cloud databases (Supabase, PlanetScale, etc.)
• GitHub-hosted repos where Claude is driving PRs, CI, and deploys
• MCP servers that connect to external APIs

For these projects, the "desktop" is just a UI shell — all the actual work happens over the network. There's no technical reason the mobile app can't participate in those sessions.

Expected behavior

• Projects visible in the desktop sidebar should optionally be accessible from the Claude Code mobile app
• For remote/cloud-backed projects, the full project context (MCP tools, system prompts, memory, conversation history) should be available on mobile
• Ideally: a "remote-compatible" badge or indicator on projects that don't require local desktop resources, so users know which ones are safe to open on mobile

Why this matters

Right now, switching to mobile means losing all project context and starting a blank conversation. For remote-first projects this is a significant unnecessary limitation — especially for quick edits, monitoring CI, checking deployments, or responding to alerts on the go.

Out of scope (acknowledged)

Projects that depend on local file system access, locally running dev servers, or desktop-only tools would reasonably remain desktop-only. This request is specifically about projects where that dependency doesn't exist.