Instead of reading in STDOUT of `ssh`, redirect the output to a file:

    $backup_command = "backup.pl >! output123.txt";
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Also, when "killing" the process, you may wish to send a SIGTERM first before SIGKILL (like kill -9). This gives your $backup_command a change to catch and handle the signal to end its operation (i.e., flushing out its output buffer).

See perlipc on how to catch signals.
See Suffering from Buffering on how to flush output.