no warnings 'regexp'; ($str0 ^ $str1) =~ /^((??{chr(0)})*)/;
$commonLen = $+[0];
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Two things

• AFAICT @+ doesn't require capturing parens,
• unless you use single quotes as delimiters, regexen obey the same substituon rules as double quoted strings.

Thus

($str0 ^ $str1) =~ /^\0*)/;
[download]

should suffice and it shouldn't even be necessary to initialize $commonLen to zero:

#!/usr/bin/perl -l

use strict;
use warnings;

sub clen {
    ($_[0] ^ $_[1]) =~ /^\0*/;
    $+[0];
}

print +(clen split) while <DATA>;
__END__
aaaa bbbb
aaaa abbb
aaaa aabb
aaaa aaab
aaaa aaaa
[download]