My two cents:

my $data; $data .= $_ while (<>);
for ( ($data =~ m/<([^\/>]+)/sg) ) {
  push(@opened, $_) if ! ($data =~ s/<\/$_>//s);
}
print "Opened: ", join(' - ', @opened), "\n";
print "Closed: ", join(' - ', ($data =~ m/<\/([^>]+)/sg) ), "\n";
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Obviously not for complex tags - :-)
Best,
  -Adam