claude-in-chrome MCP tools fail when Claude desktop app is also installed

When both the Claude desktop app and Claude Code CLI are installed on the same machine, the Chrome extension connects to the desktop
app's native host instead of Claude Code's, causing all mcp__claude-in-chrome__* tools to return "Browser extension is not connected."

Root cause:

• The Chrome extension uses com.anthropic.claude_browser_extension as the native messaging host name
• The desktop app registers com.anthropic.claude_browser_extension → /Applications/Claude.app/Contents/Helpers/chrome-native-host
• Claude Code registers a different name: com.anthropic.claude_code_browser_extension → ~/.claude/chrome/chrome-native-host
• Since the extension only calls com.anthropic.claude_browser_extension, it always connects to the desktop app's native host, not Claude

Code's

Steps to reproduce:

1. Install Claude desktop app
2. Install Claude Code CLI
3. Run claude --chrome in a project
4. Attempt to use any mcp__claude-in-chrome__* tool
5. All calls fail with "Browser extension is not connected"

Expected: Claude Code should work alongside the desktop app, or the extension should be able to disambiguate

Environment: macOS 15.2, Chrome, Claude desktop app + Claude Code CLI both installed