Untested:

foreach my $pattern (@patterns) {
    for (my $i = 0; $i < length($string); $i += 3) {
        push @matches, ${^MATCH}
              if substr($string, $i) =~ /^$pattern/p;
    }
}
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It shouldn't be hard to adapt if you want the offsets (as your original code does). It does capture overlapping matches, although with the given patterns, no overlap is possible.