Depending on what you're doing, you might either make sure that your variables are gone before global destruction (by, for instance, making them all in an inner lexical scope), or maybe to just ignore doing certain cleanup during global destruction. You can tell if you're in global destruction using this trick:

package MyStuff;
my $ending = 0;
sub END { $ending = 1 }
sub DESTROY {
   return if $ending;
   # more cleanup here
}
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Of course, it depends on what kind of cleanup you need to do; writing to files, etc. would not be something you'd want to skip, while you could probably skip closing them.