You could build up a reverse hash with arrayrefs, then count the length of those:

my %forward_hash = (... your hash...);
## accumulate the reverse mapping
my %reverse_hash;
while (my($k, $v) = each %forward_hash)) {
  push @{$reverse_hash{$v}}, $k;
}
## now see if there are dups:
while (my($k, $v) = each %reverse_hash)) {
  my @vals = @$v;
  next unless @vals > 1; # skip single values
  print "value $k is shared by keys @vals\n";
}
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