This can be solved pretty easily using the Tree::Suffix module.
Example:
#!/usr/bin/env perl
use Modern::Perl;
use Bio::SeqIO;
use Tree::Suffix;
my $seqio = Bio::SeqIO->new(
-file => "fasta_sequences.txt",
-format => "Fasta");
my @seqs;
while (my $seqobj = $seqio->next_seq) {
push @seqs, $seqobj->seq;
}
my $tree = Tree::Suffix->new(@seqs);
my @lcss = $tree->lcs;
say $_ for @lcss;