Taking in mind that while (<F1>) is short for while (defined ($_ = <F1>)), I would write that as:

while (defined(my $f1 = <F1>) && defined(my $f2 = <F2>)) {
   ...
}
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This assumes you want to stop reading if you reach the end of either file.

Oh, and you need to change how you're opening the file. You now open the files for writing.