Bash run_in_background tasks killed at turn boundary on MSYS2 MINGW64, and reported status does not match the process

### Summary

On an MSYS2 MINGW64 shell, `Bash` tool calls launched with
`run_in_background: true` are terminated at the turn boundary after launch, and
the task-completion notification does not reflect what actually happened to the
process. Observed independently by two sessions on the same host.

### Environment

- Claude Code 2.1.239
- Windows 11 Home 10.0.26200
- `uname -a`: `MINGW64_NT-10.0-26200 3.6.10-8fbd9808.x86_64 2026-07-31 07:26 UTC x86_64 Msys`
- Shell: MSYS2 MINGW64, installed at `C:\msys64` — the MSYS2 distribution, **not** the Git Bash bundled with Git for Windows. Reproducing under Git for Windows may behave differently; this report is from MSYS2. Via the `Bash` tool, not PowerShell.

### Steps to reproduce

1. Start a long-running job that writes to a file, e.g.
`bash long-suite.sh > /tmp/out.txt 2>&1` with `run_in_background: true`.
2. Also launch a waiter whose body is dominated by `sleep`, e.g.
`until grep -q DONE /tmp/out.txt; do sleep 30; done`, with
`run_in_background: true`.
3. Finish the turn (produce any assistant message) and continue the session.

### Expected

The background task keeps running until it finishes or is explicitly stopped,
and its completion notification reports its real outcome.

### Actual

**A. Background tasks are killed at the turn boundary.** 13 waiter tasks of the
shape in step 2 (with `sleep` values from 30 to 300 seconds) each came back
`status: killed` at the first turn boundary after launch, with empty output.
Task ids from one session: `btbhyisr2`, `birrx06qk`, `bffou1ruw`, `bsdmdvfzz`,
`ba9du180o`, `blwlx45dr`, `b4npuftvc`, `bvghq9d5i`, `boz63alzo`, `bu8ont2tk`,
`b8yas9ern`, `b7y0etxp5`, `bpbwvhy9v`.

**B. The reported status does not track the process.** In both directions:

- Task `bfr9lvua5` (`bash test/run.sh`, a ~1h test suite) was reported
`killed`, yet its child processes kept running and kept appending to the
redirect target for another ~20 minutes — confirmed with `ps -ef` and a
growing `wc -l` on the output file.
- Tasks `bri6kw5e2` and `bioznef6s` were reported `completed (exit code 0)`
for runs that had been terminated by hand with `kill -9` before the
notification arrived.

(B) is the more damaging half: a status line that says `completed` for a run
that never finished, or `killed` for one still running, cannot be used to decide
whether a validation run actually produced its result.

### Workaround

Launching the same work detached from an ordinary **foreground** tool call
survives every turn boundary:

```bash
nohup ./run-batch.sh > /tmp/batch.log 2>&1 & disown
```

A 34-suite sequential test batch ran to completion this way across ~15 turns on
the same host where `run_in_background` was killed every time. The other session
reports the same workaround, with a fifo used to block on the result.

### Why it matters

On this host class the background mechanism is the only option for a long
command, and an agent that relies on it either loses the run or — worse —
records a green result it never observed.

Start by reproducing the Bash tool's run_in_background behavior on MSYS2 MINGW64 with the long-running commands and turn boundary described in the issue. Trace the background-task lifecycle and completion notification, then verify that tasks survive the boundary and that reported status matches the actual process outcome.

索引モデルが issue の本文から書いたものです。