Frankly I'd tackle this by sorting each file using the Unix sort utility and then scanning in parallel merge to filter them.

The scanning logic would look like this (untested):

#! /usr/bin/perl -w
use strict;

unless (2 == @ARGV) {
  die "Usage: $0 file1 file2";
}

open(FILE1, "<", $ARGV[0]) or die "Can't read '$ARGV[0]': $!";
open(FILE2, "<", $ARGV[1]) or die "Can't read '$ARGV[1]': $!";

my $string1 = <FILE1>;
my $string2 = <FILE2>;

while (not $is_end) {
  if ($string1 lt $string2) {
    $string1 = <FILE1> or last;
  }
  elsif ($string1 gt $string2) {
    $string2 = <FILE2> or last;
  }
  else {
    print $string1;
    $string1 = <FILE1> or last;
    $string2 = <FILE2> or last;
  }
}
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