This might work as well by inspecting the %constant::declared hash:

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

use constant { FOO => 1, BAR => 2, BAZ => 1 };

my %val_to_name;

{
   no strict 'refs';
   foreach my $con (keys(%constant::declared)) {
      push @{$val_to_name{$con->()}}, $con;
   }
}

my $thing = FOO;
print "$thing defined by: " . join(" and ", @{$val_to_name{$thing}})."
+\n";
[download]

Used a HoA to allow for multiple constants having the same value.