[BUG] Claude Code web inconsistently provides GitHub MCP tools for private repos

Preflight Checklist

• [x] I have searched existing issues and this hasn't been reported yet
• [x] This is a single bug report (please file separate reports for different bugs)
• [x] I am using the latest version of Claude Code

What's Wrong?

• The system prompt tells Claude to use mcp__github__* tools for GitHub interactions
• These tools are not reliably injected into web sessions — sometimes they're there, sometimes they're not

Claude Code web already has GitHub OAuth (it can clone private repos), so the auth exists

• When the tools aren't available, Claude tells the user it "can't access GitHub issues in private repos" — which is misleading since the capability exists, it's just not loaded
• No repo-level workaround exists because web sessions have no persistent env vars or local config

gh CLI is also not available in web sessions

What Should Happen?

If the system prompt references mcp__github__ tools, they should be available in every session. The platform already has the GitHub OAuth credentials.

Actual: Tools are intermittently missing. No repo-level workaround exists — no gh CLI, no persistent env vars, WebFetch fails on authenticated URLs.

Error Messages/Logs

Steps to Reproduce

1. Open Claude Code on the web with a private GitHub repo (e.g., timescale/tiger-den)
2. Ask Claude to read a GitHub issue: "Read issue #70"
3. In some sessions, mcp__github__ tools are available and it works perfectly
4. In other sessions, the same ToolSearch returns zero results and Claude says it cannot access issues
5. The tools can even appear/disappear mid-session — in the session where I'm filing this bug, the tools were missing for the first half of the conversation, then suddenly showed up later

Claude Model

None

Is this a regression?

I don't know

Last Working Version

_No response_

Claude Code Version

Claude code on the web

Platform

Anthropic API

Operating System

macOS

Terminal/Shell

Other

Additional Information

_No response_