No, it's not just the fork (which would be weird anyway, as a fork won't change $$ in the parent anyway):

#!/usr/bin/perl -T
use strict;
use warnings;

$ENV{PATH} = '/usr/bin:/usr/local/bin:/bin';
$ENV{ENV} = '';

fork || exit;

my $temp_fails = [
                   # `mktemp /tmp/temp.$$.XXXXXX`,
                   `mktemp /tmp/temp2.$$.XXXXXX`,
                  ];

print "Success\n";
 
__END__
Success
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Abigail