You could do it this way. I'm using a manual position rather than utilizing the default characteristics of /g. I advance the position by one character each time through the loop. And if I get a match, I advance it by one plus the offset of the beginning of that match. It probably could be golfed down a lot (for instance, the use of $sstr is unnecessary; substr can be bound directly to the regexp. And there is probably not really a need for the "else" condition in the loop, but I just wanted to cover all bases and make it as clear as possible. Here it is...

use strict;
use warnings;
my $string = "aabbaaaabbaaabbbbabaabbbaaaa";
my @a;
my $position = 0;
while ( $position < length $string ) {
    my $sstr = substr($string,$position);
    if ( $sstr =~ /(aa)/ ) {
        push @a, $1;
        $position+=$-[0]+1;
    } else {
        $position++;
    }
}
print join("-", @a), "\n";
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