Are the lines in the two files sorted by the first three characters? Your sample seems to indicate so. If not, the problem is harder - if there's a mismatch, you don't know on what side the information is missing.

#!/usr/bin/perl
use strict;
use warnings;

open my $A,   '<', 'A' or die $!;
open my $B,   '<', 'B' or die $!;
open my $NEW, '>', 'C' or die $!;

my ($lineA, $lineB);
while (! eof $A && ! eof $B) {
    chomp($lineA //= <$A>);
    chomp($lineB //= <$B>);
    my ($keyA, $keyB) = map { substr $_, 0, 3 } $lineA, $lineB;

    if ($keyA eq $keyB) {
        print "$lineA\t$lineB\n";
        undef $lineA;
        undef $lineB;

    } elsif ($keyA lt $keyB) {
        print "$lineA\n";
        print {$NEW} "$lineA\n";
        undef $lineA;

    } else {
        print "\t$lineB\n";
        print {$NEW} "$lineB\n";
        undef $lineB;
    }
}
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