I built a project achieving similar goals. You launch a web server then connect to it using either browser or Android app, then create a session to talk to Claude Code. The sessions are synchronized in real time across all devices and automatically saved to disk and continued when server restarts. Recently I've added features to schedule tasks in the future and to assemble agent teams. The project is mostly vibe-coded with Opus 4.6 with few supervision beyond trying its functionalities out.
Project is at http://github.com/vincent-163/claude-code-multi/. Can be installed easily with nodejs.
Please provide feedbacks and suggestions!