You probably want to read each file explicitly and use a "last" (or a breaking variable) to exit the loop when done. something like:

my $more_to_read = 1;
while($more_to_read){
  $more_to_read = 0 if(eof(F1) && eof(F2));
  my $f1 = <F1>;
  my $f2 = <F2>;
  #some processing here
}
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UPDATE: You should probably use strict and warnings, it will give you better diagnostic info on what actually went wrong