No tool available to enumerate running background bash tasks

Summary

When Claude launches background bash tasks via Bash(run_in_background=True), there is no tool to list or discover them. TaskList is unrelated (it only shows TaskCreate items). TaskStop and TaskOutput both require a task ID. If the task ID is lost — e.g. through a long conversation or context compaction — the process becomes unmanageable from Claude's side.

Reproduction

1. Run Bash(command="sleep 99999", run_in_background=True) — returns a task ID
2. There is no tool call that can rediscover this task or its ID
3. TaskStop and TaskOutput work correctly when given the ID directly

Observed behavior

• Running background tasks are visible to the user in the Claude Code UI
• Claude has no corresponding API to enumerate them
• The only recovery path is scrolling back through the conversation to find the task ID

Expected behavior

A tool (e.g. an extension to TaskList, or a dedicated TaskListBackground) that surfaces running background bash tasks with their IDs, commands, and status — mirroring what the UI already shows.

Additional context

• TaskStop and TaskOutput work correctly by ID — the missing piece is discovery
• Non-terminating background commands (e.g. log followers) can accumulate silently across a session with no way for Claude to detect or clean them up