Something like /pattern(?{ code } )/g (see also (??{...})

But in your simple case an iterator with while (/pattern/g) {code} is certainly better style (readability, maintainability, not an experimental feature!¹)

PS: plz use code-tags

update

  DB<123> $str = join "","a".."j"
 => "abcdefghij"

  DB<124> $x=0;$str =~ /(\w)(?{print "match ",++$x," $1 \n"})/g
match 1 a 
match 2 b 
match 3 c 
match 4 d 
match 5 e 
match 6 f 
match 7 g 
match 8 h 
match 9 i 
match 10 j 
 => ("a", "b", "c", "d", "e", "f", "g", "h", "i", "j")

  DB<125> $x=0; while ( $str =~ /(\w)/g ) {print "match ",++$x," $1 \n
+"}
match 1 a 
match 2 b 
match 3 c 
match 4 d 
match 5 e 
match 6 f 
match 7 g 
match 8 h 
match 9 i 
match 10 j 
 => ""
[download]

¹) perlre 5.10-5.16

       "(?{ code })"
                 WARNING: This extended regular expression feature is
                 considered experimental, and may be changed without n
+otice.
                 Code executed that has side effects may not perform
                 identically from version to version due to the effect
+ of
                 future optimisations in the regex engine.
[download]

$count = $var =~ tr/s/s/;
[download]

I was wondering if there is a capability in Perl to register a call back to the regex engine and call that registered function each time a particular match is made.

e  Evaluate the right side as an expression.
ee Evaluate the right side as a string then eval the result.

[I was confused by the term "arbitrary code", which is normally used to refer to how s///ee can execute code found in data. Ignore this post.]