Unless the individual values get very large, I think the easiest would be to factor them:

  sub combine_factors {
    my $list = shift;
    my $factors = {};
    for (@$list) {
      for (factors($_)) {
        ++$factors->{$_};
      }
    }
    $factors;
  }

  my $fa = combine_factors(\@a);
  my $fb = combine_factors(\@b);
  ($fa->{$_} ||= 0) -= $fb->{$_} for keys %$fb;
  my @c = map +($_) x $fa->{$_}, grep $fa->{$_} > 0, keys %$fa;
  my @d = map +($_) x -$fa->{$_}, grep $fa->{$_} < 0, keys %$fa;
[download]

This has the disadvantage that it will leave a list of primes: @c == (3, 5, 11), @d == (2, 2, 2, 23). I'm not sure how you'd retain the original composites (or parts thereof), nor how you'd define which composites to keep, so I chose to ignore that aspect of it.

Hugo