When you hear "unique", think "hash". In this case, you need to hash headers by sequences:

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

my $fasta = << '__FASTA__';
>gi1 cds
ATG fun
>gi2 cds
ATG fun
>gi3 cds
GGG fun
__FASTA__

my @seq_with_hdr = split /\n>/, $fasta;
$seq_with_hdr[0] =~ s/^>//;

my %hdr_by_seq;

for (@seq_with_hdr) {
    my ($hdr, $seq) = split /\n/;
    $hdr_by_seq{$seq} = $hdr;
}

for my $seq (keys %hdr_by_seq) {
    print ">$hdr_by_seq{$seq}\n$seq\n"
}
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Note that whitespace is not ignored in the data. There was a space after one of "ATG FUN" sequences which makes it different to the same sequence without the trailing space. I removed the space in my code.