Claude Code Chrome integration fails to connect when Claude Desktop (Cowork) is also installed

Description

Claude Code's Chrome browser integration (tabs_context_mcp) never connects when Claude Desktop is also installed
with the Cowork/browser feature enabled. The extension remains in a disconnected state even with Claude Desktop fully
closed.

## Environment

• Claude Code version: 2.1.59
• Extension version: 1.0.56
• Extension ID: fcoeoabgfenejglbffodgkkbkcdhcgfn
• OS: macOS (Darwin 25.3.0)
• Chrome: latest

## Root Cause (Investigated)
Both Claude Desktop and Claude Code register native messaging hosts for the same extension ID:

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

Claude Desktop force-installs the extension via Chrome's External Extensions policy (~/Library/Application
Support/Google/Chrome/External Extensions/fcoeoabgfenejglbffodgkkbkcdhcgfn.json), causing Chrome to label it "Added
by a third-party". The extension's service worker opens the Desktop Cowork side panel UI but never initiates a
connection to Claude Code's native host.

## Steps to Reproduce

1. Install Claude Desktop with Cowork/browser integration enabled
2. Install Claude Code and set up Chrome integration
3. Close Claude Desktop completely
4. Attempt to use Claude Code's Chrome tools — connection always fails

## What Was Tried

• Full Chrome restarts (multiple)
• Removing and reinstalling the extension
• Verified native messaging host config is correct with matching extension ID
• Verified Claude Code binary is functional (native host creates socket successfully)
• Verified account match (same email in Claude Code auth and claude.ai in Chrome)
• Temporarily hiding Desktop's native messaging host file — did not resolve it

## Expected Behavior
Claude Code's Chrome integration should work independently of whether Claude Desktop is installed.

## Actual Behavior
tabs_context_mcp always returns: "Browser extension is not connected"

The extension only opens the Desktop Cowork side panel UI and does not connect to Claude Code's native messaging host.