Just for fun :)
It only takes one s/// (with a little extra state)

#!/usr/bin/perl

use strict; # https://perlmonks.org/?node_id=11144309
use warnings;

my $str = 'word1 <tag0> word2 <tag1>word3 word4</tag1> word5 </tag0> w
+ord6 <tag2>word7 word8</tag2>word9 <tag3>word10</tag3> word11';

print parsestringwithtags( $str, 'tag0', 'tag1' ), "\n";
print parsestringwithtags( $str, 'tag3' ), "\n";
print parsestringwithtags( $str, 'tag1', 'tag2', 'tag3' ), "\n";

sub parsestringwithtags
  {
  local $_ = shift;
  my %tags = map { $_, 1 } @_;
  my $active = 1;
  my @state = ['', 1];
  s{ <(/?)(\w+)> | ([^<>]+) | ([<>]) }{
    $4
      ? die "rogue angle bracket $4 at $-[4]"
      : $2                           # tag
        ? $1                         # it is an end tag
          ? do { $state[-1][0] eq $2 # matches current tag
            ? do { $active = (pop @state)->[1]; '' }
            : die "mismatched tags $state[-1][0] vs $2" }
          : do { push @state, [$2, $active]; $tags{$2} or $active = 0;
+ '' }
        : $3 x $active               # non-tag only if active
    }gex;
  @state > 1 and die "missing close tag for $state[-1][0]";
  return $_;
  }
[download]

Outputs:

word1  word2 word3 word4 word5  word6 word9  word11
word1  word6 word9 word10 word11
word1  word6 word7 word8word9 word10 word11
[download]