Pretty much what Ikegami said on SO. You can see the order that the twigs and residual tags are being processed by modfying the code as follows:

use XML::Twig;
use IO::Tee;
use feature 'say';

open my $frufile, '>', 'fruit.xml' or die "fruit $!";
open my $vegfile, '>', 'veg.xml' or die "veg $!";

my $tee = IO::Tee->new($frufile, $vegfile);
select $tee;

my $twig=XML::Twig->new(
    twig_handlers => {
        thing  => \&magic,
        _default_  => sub { 
            print '_default_ for '.$_->name." [[[";
            $_[0]->flush($tee); #default filehandle = tee
            say "]]]";
            1; 
        },
    },
    pretty_print => 'none',
    empty_tags   => 'normal',
);

$twig->parse( *DATA );

sub magic {
    my ($thing, $element) = @_;
    print "magic for ". $element->{att}{type}." [[[";
    for ($element->{att}{type}) {
            if (/fruit/) {
                $thing->flush($frufile);
            } elsif (/vegetable/) {
                $thing->flush($vegfile);
            } else {
                $thing->purge;
            }
    }
    
    say "]]]";
    1;
}

__DATA__
<batch>
  <header>
    <foo>1</foo>
    <bar>2</bar>
    <baz>3</baz>
  </header>
  <thing_list>
  <thing type="fruit"     >Im an apple!</thing>
  <thing type="city"      >Toronto</thing>
  <thing type="vegetable" >Im a carrot!</thing>
  <thing type="city"      >Melrose</thing>
  <thing type="vegetable" >Im a potato!</thing>
  <thing type="fruit"     >Im a pear!</thing>
  <thing type="vegetable" >Im a pickle!</thing>
  <thing type="city"      >Patna</thing>
  <thing type="fruit"     >Im a banana!</thing>
  <thing type="vegetable" >Im an eggplant!</thing>
  <thing type="city"      >Taumatawhakatangihangakoauauotamateaturipuk
+akapikimaungahoronukupokaiwhenuakitanatahu</thing>
  </thing_list>
  <trailer>
    <chrzaszcz>A</chrzaszcz>
    <zdzblo>B</zdzblo>
  </trailer>
</batch>
[download]

Fruit.xml:

_default_ for foo[[[<batch><header><foo>1</foo>]]]
_default_ for bar[[[<bar>2</bar>]]]
_default_ for baz[[[<baz>3</baz>]]]
_default_ for header[[[</header>]]]
magic for fruit [[[<thing_list><thing type="fruit">Im an apple!</thing
+>]]]
magic for city [[[]]]
magic for vegetable [[[]]]
magic for city [[[]]]
magic for vegetable [[[]]]
magic for fruit [[[<thing type="fruit">Im a pear!</thing>]]]
magic for vegetable [[[]]]
magic for city [[[]]]
magic for fruit [[[<thing type="fruit">Im a banana!</thing>]]]
magic for vegetable [[[]]]
magic for city [[[]]]
_default_ for thing_list[[[</thing_list>]]]
_default_ for chrzaszcz[[[<trailer><chrzaszcz>A</chrzaszcz>]]]
_default_ for zdzblo[[[<zdzblo>B</zdzblo>]]]
_default_ for trailer[[[</trailer>]]]
_default_ for batch[[[</batch>]]]
[download]

Veg.xml

_default_ for foo[[[<batch><header><foo>1</foo>]]]
_default_ for bar[[[<bar>2</bar>]]]
_default_ for baz[[[<baz>3</baz>]]]
_default_ for header[[[</header>]]]
magic for fruit [[[]]]
magic for city [[[]]]
magic for vegetable [[[<thing type="vegetable">Im a carrot!</thing>]]]
magic for city [[[]]]
magic for vegetable [[[<thing type="vegetable">Im a potato!</thing>]]]
magic for fruit [[[]]]
magic for vegetable [[[<thing type="vegetable">Im a pickle!</thing>]]]
magic for city [[[]]]
magic for fruit [[[]]]
magic for vegetable [[[<thing type="vegetable">Im an eggplant!</thing>
+]]]
magic for city [[[]]]
_default_ for thing_list[[[</thing_list>]]]
_default_ for chrzaszcz[[[<trailer><chrzaszcz>A</chrzaszcz>]]]
_default_ for zdzblo[[[<zdzblo>B</zdzblo>]]]
_default_ for trailer[[[</trailer>]]]
_default_ for batch[[[</batch>]]]
[download]

Notice how the "<thing>" handler gets called after the tag is closed and flushes any other previous, unprocessed tag information before it.