might Text::Balanced be what yuo are looking for?

PS

use strict;
use warnings;
use Text::Balanced qw(extract_tagged);

my $str = '<tag0> word2 <tag1>word3 word4</tag1> word5 </tag0> word6 <
+tag2>word7 word8</tag2> word9 <tag3>word10</tag3>';

# this leads to errors
# my $str = 'BEFORE <tag0> word2 <tag1>word3 word4</tag1> word5 </tag0
+> word6 <tag2>word7 word8</tag2> word9 <tag3>word10</tag3> AFTER';

my @res = extract_tagged( $str , '<tag0>', '</tag0>');

foreach (qw(extracted remainder prefix opening included closing)){
    
    my $res = shift  @res // 'NA';
    print "$_:\t$res\n";
    
}
__END__

extracted:      <tag0> word2 <tag1>word3 word4</tag1> word5 </tag0>
remainder:       word6 <tag2>word7 word8</tag2> word9 <tag3>word10</ta
+g3>
prefix:
opening:        <tag0>
included:        word2 <tag1>word3 word4</tag1> word5
closing:        </tag0>
[download]

L*