Hi, here is one solution. I'm using Path::Tiny for file handling. Also localizing the special variable $/ (the input record separator), as you hinted at.

use strict; use warnings; use feature 'say';
use Path::Tiny;

local $/ = '>';

my $fh = path('./foo.fasta')->openr;

while ( my $paragraph = <$fh> ) {
    chomp $paragraph;
    my @lines = split /\n/, $paragraph or next;
    my ( $identifier, $string );
    for my $line ( @lines ) {
        if ( $line =~ /(sequence\d+)/ ) {
            $identifier = $1;
        } else {
            $string .= $line;
        }
    }
    say "$identifier\t$string";
}

__END__
[download]

I used the following input file:

>sequence1
ACTCCCCGTGCGCGCCCGGCCCGTAGCGTCCTCGTCGCCGCCCCTCGTCTCGCAGCCGCAGCCCGCGTGG
ACGCTCTCGCCTGAGCGCCGCGGACTAGCCCGGGTGGCC

> sequence2
CAGTCCGGCAGCGCCGGGGTTAAGCGGCCCAAGTAAACGTAGCGCAGCGATCGGCGCCGGAGATTCGCGA
ACCCGACACTCCGCGCCGCCCGCCGGCCAGGACCCGCGGCGCGATCGCGGCGCCGCGCTACAGCCAGCCT
CACTGGCGCGCGGGCGAGCGCACGGGCGCTC

>randomstuff sequence3
CACGACAGGCCCGCTGAGGCTTGTGCCAGACCTTGGAAACCTCAGGTATATACCTTTCCAGACGCGGGAT
CTCCCCTCCCC

> sequence4 blahblah
CAGCAGACATCTGAATGAAGAAGAGGGTGCCAGCGGGTATGAGGAGTGCATTATCGTTAATGGGAACTTC
AGTGACCAGTCCTCAGACACGAAGGATGCTCCCTCACCCCCAGTCTTGGAGGCAATCTGCACAGAGCCAG
TCTGCACACC
[download]

and got the following output:

$ perl foo.pl
sequence1    ACTCCCCGTGCGCGCCCGGCCCGTAGCGTCCTCGTCGCCGCCCCTCGTCTCGCAGCC
+GCAGCCCGCGTGGACGCTCTCGCCTGAGCGCCGCGGACTAGCCCGGGTGGCC
sequence2    CAGTCCGGCAGCGCCGGGGTTAAGCGGCCCAAGTAAACGTAGCGCAGCGATCGGCGC
+CGGAGATTCGCGAACCCGACACTCCGCGCCGCCCGCCGGCCAGGACCCGCGGCGCGATCGCGGCGCCGC
+GCTACAGCCAGCCTCACTGGCGCGCGGGCGAGCGCACGGGCGCTC
sequence3    CACGACAGGCCCGCTGAGGCTTGTGCCAGACCTTGGAAACCTCAGGTATATACCTTT
+CCAGACGCGGGATCTCCCCTCCCC
sequence4    CAGCAGACATCTGAATGAAGAAGAGGGTGCCAGCGGGTATGAGGAGTGCATTATCGT
+TAATGGGAACTTCAGTGACCAGTCCTCAGACACGAAGGATGCTCCCTCACCCCCAGTCTTGGAGGCAAT
+CTGCACAGAGCCAGTCTGCACACC
[download]

Hope this helps!