If I'm understanding what you are going for, I'd use a hash of arrays of hashes. That way you keep all of the data and can easily find 'duplicated keys':

#!/usr/bin/env perl
use strict;
use warnings;
use feature 'say';
use Data::Printer;

my %hash = (
    alpha1 => [
        { beta => { gamma => 'theta', delta => 'lambda', } },
        { beta => { gamma => 'zeta', } },
    ],
    alpha2 => [
        { beta => { gamma => 'theta', delta => 'lambda', } },
    ],
);

say "## Original state:";
check_duplication();

push @{ $hash{alpha2} }, { beta => { gamma => 'theta', } };

say "## After pushing another element onto 'alpha2':";
check_duplication();

sub check_duplication {
    for ( keys %hash ) {
        if ( scalar @{ $hash{$_} } > 1 ) {
            say "$_ has a duplicate.";
        }
        else {
            say "$_ is unique.";
        }
    }
}

p %hash;
[download]

OUTPUT:

{
    alpha1   [
        [0] {
            beta   {
                delta   "lambda",
                gamma   "theta"
            }
        },
        [1] {
            beta   {
                gamma   "zeta"
            }
        }
    ],
    alpha2   [
        [0] {
            beta   {
                delta   "lambda",
                gamma   "theta"
            }
        },
        [1] {
            beta   {
                gamma   "theta"
            }
        }
    ]
}
## Original state:
alpha2 is unique.
alpha1 has a duplicate.
## After pushing another element onto 'alpha2':
alpha2 has a duplicate.
alpha1 has a duplicate.
[download]