I recommend that you use the perlbug command line tool to ask p5p to explain the outcome of the following cleaned up version of your program. Include the output.

use strict;
use warnings;
use feature qw( say );

use Data::Dumper qw( Dumper );

BEGIN {
   package str;

   use strict;
   use warnings;
   use feature qw( say );

   use overload
      '""' => \&stringify,
      '.'  => \&concat;

   sub new { my $class = shift; bless([ @_ ], $class) }

   sub stringify { join '', @{ $_[0] } }

   sub concat {
      my ($s1, $s2, $inverted) = @_;
      return ref($s1)->new( $inverted ? ($s2,$s1) : ($s1,$s2) );
   }

   $INC{"str.pm"} = 1;
}

sub _dump {
   local $Data::Dumper::Indent = 0;
   local $Data::Dumper::Terse = 1;
   local $Data::Dumper::Sortkeys = 1;
   return Dumper(@_);
}

{
   my $str = str->new('foo');

   my $good = "a $str a";
   say "good: "._dump($good);

   my $bad; $bad = "a $str a";
   say "bad: "._dump($bad);
}
[download]

Output:

good: bless( [bless( ['a ',bless( ['foo'], 'str' )], 'str' ),' a'], 's
+tr' )
bad: 'a foo a'
[download]