[Bug] Background process reminders persist after /clear command for completed processes

Bug Description
Claude Code's background process monitoring generates persistent system reminders for
completed Bash processes, even after using /clear to start a fresh conversation. When running
commands like echo "..." | codex exec that create background shells, the system continues
showing reminders (e.g., "Background process fc6729 has new output available") indefinitely.
The reminders claim processes have status "running" but attempting to kill them returns
"status: completed". This creates a feedback loop where the assistant is prompted to check
old output repeatedly, consuming context tokens unnecessarily. Expected behavior: /clear
should purge all background process state and stop generating reminders for completed
processes.
Current workaround -> "fixes" the issue: Manually restart terminal/IDE, but this shouldn't be required.
The issue appears specific to long-running streaming processes (Codex analyses) where the
system doesn't properly recognize completion state.