To do what I think you want from:

while ($line1 =<F1> || $line2= <F2>){
  foo($line1,$line2);
}
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I suggest something along the lines of (i.e. untested):

  local $/ ;
  while (1) {
    $/ = "#" ; my $line1 = <F1> ;
    $/ = "?" ; my $line2 = <F2> ;
    last if (!defined($line1) && !defined($line2)) ;
    foo($line1, $line2) ;
  } ;
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Noting that if the files are different lengths, either $line1 or $line2 may be undefined when passed to foo. Stopping as soon as the shorter hits eof is a straightforward modification -- throwing an error if the two files are supposed to be the same length as necessary.