The regular expression s/(.+)-\1/$1/ will solve it almost. It won't satisfy (yet) your "last-match" rule

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

my %exampledata= (
    ATTTA  => 'TTTAA',
    ATGTA  => 'ATGTA',
    ATGATG => 'ATGATG',
);

while (my($s1, $s2)= each %exampledata) {
    print $s1,'+',$s2,' => ',merge_data($s1, $s2),"\n";
}
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sub merge_data {
    local($_)= join("-", @_);
    s/(.+)-\1/$1/;
    return $_;
}
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Update: This will take care of the rule (in readmore)

sub merge_data {
    local($_)= join("-", @_);
    my($m_str)= $_;
    if (s/(.+)-\1/$1/) {
        my $first_result= $_;
        my $chop_off= length($_[0])-length($1)+1;
        $_= substr($m_str, $chop_off);
        if (s/(.+)-\1/$1/) {
            return substr($m_str, 0, $chop_off) . $_;
        }
        return $first_result;
    }
    return $_;
}
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Update: This will take care of example 2 by inserting an arbitrary rule.

Rules are now: Find all matches and take a) the second best match that's longer than 1 character b) the best match

sub merge_data {
    local($_)= join("-", @_);
    my($m_str)= $_;
    if (s/(.+)-\1/$1/) {
        my $first_result= $_;
        my $chop_off= length($_[0])-length($1)+1;
        $_= substr($m_str, $chop_off);
        if (s/(.{2,})-\1/$1/) {
            return substr($m_str, 0, $chop_off) . $_;
        }
        return $first_result;
    }
    return $_;
}
[download]