Here is another technique for your toolbox. I don't see the need to separate the identifier or create $union.

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


while (my $line = <DATA>)
{
    process_record($line) if $line=~ /^>/;
}

sub process_record
{
    my $line = shift;
    $line =~ s/\s*$//;  #or chomp $line
    print "$line\t";
    
    while (defined ($line=<DATA>) and $line !~ /^\s*$/)
    {
        $line =~ s/\s*$//;  #or chomp $line
        print $line;
    }
    print "\n";
}

=Prints:
>sequence1    ACTCCCCGTGCGCGCCCGGCCCGTAGCGTCCTCGTCGCCGCCCCTCGTCTCGCAGC
+CGCAGCCCGCGTGGACGCTCTCGCCTGAGCGCCGCGGACTAGCCCGGGTGGCC
>sequence2    CAGTCCGGCAGCGCCGGGGTTAAGCGGCCCAAGTAAACGTAGCGCAGCGATCGGCG
+CCGGAGATTCGCGAACCCGACACTCCGCGCCGCCCGCCGGCCAGGACCCGCGGCGCGATCGCGGCGCCG
+CGCTACAGCCAGCCTCACTGGCGCGCGGGCGAGCGCACGGGCGCTC
>sequence3    CACGACAGGCCCGCTGAGGCTTGTGCCAGACCTTGGAAACCTCAGGTATATACCTT
+TCCAGACGCGGGATCTCCCCTCCCC
>sequence4    CAGCAGACATCTGAATGAAGAAGAGGGTGCCAGCGGGTATGAGGAGTGCATTATCG
+TTAATGGGAACTTCAGTGACCAGTCCTCAGACACGAAGGATGCTCCCTCACCCCCAGTCTTGGAGGCAA
+TCTGCACAGAGCCAGTCTGCACACC
=cut



__DATA__
>sequence1
ACTCCCCGTGCGCGCCCGGCCCGTAGCGTCCTCGTCGCCGCCCCTCGTCTCGCAGCCGCAGCCCGCGTGG
ACGCTCTCGCCTGAGCGCCGCGGACTAGCCCGGGTGGCC

>sequence2
CAGTCCGGCAGCGCCGGGGTTAAGCGGCCCAAGTAAACGTAGCGCAGCGATCGGCGCCGGAGATTCGCGA
ACCCGACACTCCGCGCCGCCCGCCGGCCAGGACCCGCGGCGCGATCGCGGCGCCGCGCTACAGCCAGCCT
CACTGGCGCGCGGGCGAGCGCACGGGCGCTC

>sequence3
CACGACAGGCCCGCTGAGGCTTGTGCCAGACCTTGGAAACCTCAGGTATATACCTTTCCAGACGCGGGAT
CTCCCCTCCCC

>sequence4
CAGCAGACATCTGAATGAAGAAGAGGGTGCCAGCGGGTATGAGGAGTGCATTATCGTTAATGGGAACTTC
AGTGACCAGTCCTCAGACACGAAGGATGCTCCCTCACCCCCAGTCTTGGAGGCAATCTGCACAGAGCCAG
TCTGCACACC
[download]

Update:
I missed the idea that there could be some "kruft" after the id because an example of that was not present in the example data provided. I would recommend something like this change to process_record():

    (my $id) = $line =~ /^(>\w+)/;  #allow for kruft after id
    print "$id\t";
[download]

In general, use a regex when you know what you want to extract. Use a split when you know what you want to throw away. With the above regex, it is not necessary to enumerate all of these potential non-word things like "|;(space)", etc. However, note that underscore "_" is a "word character" (any character valid in a Perl variable name is allowed for \w). If underscore could directly follow the sequence id, then of course the regex would need to change. However, I see no evidence that would be necessary.