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use strict;
use warnings;
...
my $doc = $parser->parse($xml);

$doc->printToFile("test.xml");

>perl a.pl

...
2E 30 22 20 65 6E 63 6F 64 69 6E 67 3D 22 55 54
46 2D 38 22 3F 3E 0A 3C 54 45 53 54 3E 20 E9 20
3C 2F 54 45 53 54 3E 0A

use strict;
use warnings;
...

open my $fh, ">:utf8", "test.xml" or die $!;
$doc->printToFileHandle($fh);

>perl a.pl

...
2E 30 22 20 65 6E 63 6F 64 69 6E 67 3D 22 55 54
46 2D 38 22 3F 3E 0A 3C 54 45 53 54 3E 20 C3 A9
20 3C 2F 54 45 53 54 3E 0A

open my $fh, ">:utf8", "test.xml" or die $!;
print($fh "\x{FEFF}");
$doc->printToFileHandle($fh);

...Fix the <?xml?> line...
open my $fh, ">:encoding(cp1252)", "test.xml" or die $!;
$doc->printToFileHandle($fh);