This is not very thought out or tested, probably not very efficient on large lists, but I think its interesting, and it seems to work:

#!/usr/bin/perl

use strict;
use warnings;

my @array1 = qw(dog cat rat mouse);
my @array2 = qw(dog rat mouse bird);
my @array3 = qw(cat rat fish mouse);

#my @array1= ( 1, 3, 4, 6 );
#my @array2= ( 1, 2, 4, 6 );
#my @array3= ( 1, 2, 3, 5 );

init_cmp($_) for \@array1, \@array2, \@array3;
my %animals;
undef @animals{@array1, @array2, @array3};
print join(",", keys %animals), "\n";
my @sorted = sort { my_cmp($a, $b) } keys %animals;
print join(",", @sorted), "\n";

{
my %lt;
sub init_cmp {
  my ($this, @rest) = @{$_[0]};
  while ( @rest ) {
    $lt{$this}{$rest[0]} = 1;
    $this = shift @rest;
  }
}

sub is_lt {
  my ($first, $next) = @_;
  return unless exists $lt{$first};
  return 1 if $lt{$first}{$next};
  for my $mid ( keys %{$lt{$first}} ) {
    return 1 if is_lt($mid, $next);
  }
  return;
}

sub my_cmp {
  my ($first, $next) = @_;
  return is_lt($first, $next) ? -1 : 1;
}
}
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