An untested example of a possible solution:

open(FILE_A, "< file_a.txt") or die "Cannot open file_a: $!\n";
open(FILE_B, "< file_b.txt") or die "Cannot open file_b: $!\n";

while (1) {
    my $line_a = <FILE_A>;
    my $line_b = <FILE_B>;

    last if not defined $line_a or not defined $line_b;
    
    # Processing of $line_a and $line_b...
}
close(FILE_A);
close(FILE_B);
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Arjen