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Andrew: I strongly agree with the goal that IDLE should not leave zombie processes. The background process should die if either 1) IDLE restarts the shell with a new background process, as with every edit-run cycle, or 2) IDLE dies. The desired behavior seems to be both somewhat fragile and system dependent. I do not know whether 100% compliance on every system is sensibly possible.
The issue I referred to is #12540. The problem there was worse: leaving a zombie for every shell restart on Windows. Perhaps the discussion there will give you some ideas.
Ctrl-\ does not seem to do anything on Windows. I do not know whether TaskManager 'Terminate process' corresponds to *nix SIGTERM, SIGQUIT, SIGKILL, something else, or is completely Windows specific.