[FEATURE] VS Code extension's "Web" tab should list active `claude remote-control` sessions

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 run claude remote-control as a persistent systemd service on my self-hosted VPS. The session is accessible from:

• The official Claude mobile app (Code tab) — shows the session with hostname label, lets me connect and chat
• claude.ai/code in the browser — shows the same session under Recents, full web-chat UI
• The VS Code Claude Code extension — Web tab is empty (No web sessions yet), even though I'm signed into the same claude.ai account

The Web tab in the extension only lists sessions that were created directly on claude.ai/code, not remote-control sessions hosted on my own infrastructure.

Remote-control sessions are an increasingly common setup for developers who want to keep Claude Code running on a beefier always-on machine (homelab, cloud VM, dev container) while working from a laptop or mobile device. The VS Code extension is the natural home for these sessions when working at the desktop — it would let me:

• See VPS-hosted Claude sessions alongside my local ones in one UI
• Connect to an active remote session without flipping between browser and editor
• Keep my "context" (which terminal/sessions are open) consistent across devices

The mobile app and claude.ai/code already do this. The VS Code extension is the gap.

Setup details (for reference)

• VS Code: latest stable
• Claude Code extension: anthropic.claude-code (latest)
• Claude Code CLI on VPS: 2.1.112
• VPS service: `claude remote-control
• Account plan: claude.ai Max
• VPS visible in mobile app and claude.ai/code as expected; only missing from VS Code extension

Workaround in the meantime

Open claude.ai/code in a browser window alongside VS Code. Functional but breaks the "single editor surface" workflow that the extension otherwise provides.

Proposed Solution

Add remote-control sessions to the Web tab's listing. When the user signs into the extension with their claude.ai account, the extension should fetch active remote-control instances bound to that account (the same endpoint claude.ai/code already uses to populate the Recents list) and render them in the Web tab. Clicking one should open the session in an extension webview using the same chat UI as claude.ai/code, or hand off to the browser if a webview is too complex.

If discovery of remote-control instances is a different API path than claude.ai/code sessions, the Web tab could be split into two sub-sections (Cloud Sessions / Remote Control) or a single unified list with an icon distinguishing the two.

Alternative Solutions

_No response_

Priority

Medium - Would be very helpful

Feature Category

Developer tools/SDK

Use Case Example

_No response_

Additional Context

_No response_