Like others, I think you are going to have to clarify the rules, or pass a third parameter that indicates the initial offset. This comes close to matching your examples, but it falls down on your second. If the first character of the second string matches the last of the first (char/string respectively), then the overlap of 1 will always win.

P:\test\Vector maps>p1
perl> sub merge{
    my( $s1, $s2 ) = @_;
    my $i=1;
    $i++ until substr( $s1, -$i ) eq substr( $s2, 0, $i );
    return $s1 . substr( $s2, $i );
};;
perl> print merge( 'ATTTA',  'TTTAA' );;
ATTTAA
perl> print merge( 'ATGTA', 'ATGTA' );;
ATGTATGTA
perl> print merge( 'ATGATG', 'ATGATG' );;
ATGATGATG
perl> print merge( 'ATGGTAC', 'CCGTAATG' );;
ATGGTACCGTAATG
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Your addendum about order still doesn't unambiguously resolve the requirement.