One of the rare cases where you don't want to enable warnings:

use strict;
use autodie;

open my $one, "<", "data1";
open my $two, "<", "data2";

{
    my $line1 = <$one>;
    my $line2 = <$two>;
    last unless $line1 or $line2;
    chomp $line1;
    chomp $line2;
    print "$line1\t$line2\n";
    redo;
}
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