[Bug] Background task output files persist across sessions and become stale

Bug Description
Title: Background task output files accumulate and become stale/misleading

Description:

When using background bash commands (run_in_background: true), Claude Code creates output files in /private/tmp/claude/<project-path>/tasks/. These files persist
across sessions and can contain outdated information that misleads the AI or user.

Environment Info

• Platform: darwin
• Terminal: Apple_Terminal
• Version: 2.1.17
• Feedback ID: 72c58a7c-3b8c-4307-9fef-8a76d2b69592

Errors

[{"error":"Error: ENOENT: no such file or directory, scandir '/Library/Application Support/ClaudeCode/.claude/skills'\n    at readdirSync (unknown)\n    at <anonymous> (/$bunfs/root/claude:12:1903)\n    at vW (/$bunfs/root/claude:11:35097)\n    at readdirSync (/$bunfs/root/claude:12:1864)\n    at y2R (/$bunfs/root/claude:1479:380)\n    at y2R (/$bunfs/root/claude:1479:1350)\n    at <anonymous> (/$bunfs/root/claude:1479:4440)\n    at <anonymous> (/$bunfs/root/claude:1479:5171)\n    at A (/$bunfs/root/claude:11:7244)\n    at pX8 (/$bunfs/root/claude:4784:4342)","timestamp":"2026-01-22T22:46:11.362Z"},{"error":"Error: ENOENT: no such file or directory, scandir '/Users/kavanseggie/.claude/skills'\n    at readdirSync (unknown)\n    at <anonymous> (/$bunfs/root/claude:12:1903)\n    at vW (/$bunfs/root/claude:11:35097)\n    at readdirSync (/$bunfs/root/claude:12:1864)\n    at y2R (/$bunfs/root/claude:1479:380)\n    at y2R (/$bunfs/root/claude:1479:1350)\n    at <anonymous> (/$bunfs/root/claude:1479:4483)\n    at <anonymous> (/$bunfs/root/claude:1479:5171)\n    at A (/$bunfs/root/claude:11:7244)\n    at pX8 (/$bunfs/root/claude:4784:4342)","timestamp":"2026-01-22T22:46:11.362Z"},{"error":"Error: ENOENT: no such file or directory, scandir '/Library/Application Support/ClaudeCode/.claude/skills'\n    at readdirSync (unknown)\n    at <anonymous> (/$bunfs/root/claude:12:1903)\n    at vW (/$bunfs/root/claude:11:35097)\n    at readdirSync (/$bunfs/root/claude:12:1864)\n    at y2R (/$bunfs/root/claude:1479:380)\n    at y2R (/$bunfs/root/claude:1479:1350)\n    at <anonymous> (/$bunfs/root/claude:1479:4440)\n    at <anonymous> (/$bunfs/root/claude:1479:5171)\n    at A (/$bunfs/root/claude:11:7244)\n    at pX8 (/$bunfs/root/claude:4784:4342)","timestamp":"2026-01-23T00:59:03.108Z"},{"error":"Error: ENOENT: no such file or directory, scandir '/Users/kavanseggie/.claude/skills'\n    at readdirSync (unknown)\n    at <anonymous> (/$bunfs/root/claude:12:1903)\n    at vW (/$bunfs/root/claude:11:35097)\n    at readdirSync (/$bunfs/root/claude:12:1864)\n    at y2R (/$bunfs/root/claude:1479:380)\n    at y2R (/$bunfs/root/claude:1479:1350)\n    at <anonymous> (/$bunfs/root/claude:1479:4483)\n    at <anonymous> (/$bunfs/root/claude:1479:5171)\n    at A (/$bunfs/root/claude:11:7244)\n    at pX8 (/$bunfs/root/claude:4784:4342)","timestamp":"2026-01-23T00:59:03.108Z"},{"error":"Error: EISDIR: illegal operation on a directory, read\n    at readFileSync (unknown)\n    at <anonymous> (/$bunfs/root/claude:12:582)\n    at vW (/$bunfs/root/claude:11:35097)\n    at readFileSync (/$bunfs/root/claude:12:541)\n    at od_ (/$bunfs/root/claude:5192:27565)\n    at call (/$bunfs/root/claude:1488:945)\n    at call (/$bunfs/root/claude:1484:6496)\n    at dR8 (/$bunfs/root/claude:3025:19033)\n    at processTicksAndRejections (native:7:39)","timestamp":"2026-01-23T00:59:44.809Z"},{"error":"Error: EISDIR: i…

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