Here is a solution using XML::Twig. Note that it relies on say being defined before vals (so when the handler on val is called the element say is already available.:

#!/usr/bin/perl -w -l
use strict;

use XML::Twig;

my $t= XML::Twig->new( twig_handlers 
                         => { 'val[@val1="FOO"]' 
                               => sub { print $_->parent( 'monk')->fie
+ld( 'say'); }
                            }
                     )
       ->parse( \*DATA);

__DATA__
<?xml version="1.0" encoding="ISO-8859-1"?>
<monk value="PM">
 <say>JAPH</say>
  <vals>
   <val val1="F" val2="value f">
   </val>
   <val val1="FO" val2="value fo">
   </val>
   <val val1="FOO" val2="value foo">
   </val>
   <val val1="FOOB" val2="value foob">
   </val>
   <val val1="FOOBA" val2="value fooba">
   </val>
   <val val1="FOOBAR" val2="value foobar">
   </val>
  </vals>
</monk>
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