One ugly (ugly!) way that comes immediately to my mind is to use Tk::after:

# Spawn as soon as possible when the Tk main loop runs:
$phwin->after(0, \&secondary_win);
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I'm sure there also is a Tk event that is fired once the main window has drawn itself, and maybe you can initialize your secondary window from there? I don't readily see convenient callbacks for Window events in the Tk documentation, but there should be more things than just timer events.