Chrome extension conflict: Claude Desktop and Claude Code CLI compete for same extension

Description

When both Claude Desktop and Claude Code CLI are installed, the Chrome "Claude" extension (fcoeoabgfenejglbffodgkkbkcdhcgfn) cannot connect to Claude Code's MCP server. The extension always connects to Claude Desktop's native messaging host instead.

Root Cause

Both apps register native messaging hosts for the same Chrome extension:

• Claude Desktop: com.anthropic.claude_browser_extension → /Applications/Claude.app/Contents/Helpers/chrome-native-host
• Claude Code CLI: com.anthropic.claude_code_browser_extension → ~/.claude/chrome/chrome-native-host

The Chrome extension appears hardcoded to connect to com.anthropic.claude_browser_extension (Desktop), ignoring Claude Code's native host entirely.

Steps to Reproduce

1. Have both Claude Desktop and Claude Code CLI installed
2. Install the Claude Chrome extension
3. Start Claude Code CLI
4. Try to use mcp__claude-in-chrome__tabs_context_mcp tool
5. Get "Browser extension is not connected" error

Expected Behavior

Either:

• The extension should connect to whichever app initiated the request
• Or there should be a user-facing toggle to choose which app controls the browser
• Or the apps should coordinate/share the connection

Current Workaround

Quit Claude Desktop completely and restart Chrome before using Claude Code's browser features. This is disruptive.

Environment

• macOS (Darwin 25.2.0)
• Claude Code CLI: 2.1.19
• Claude Desktop: Running
• Chrome extension: v1.0.40