In Excel::Writer::XLSX::Package::Packager there is:

sub _create_package {
    ...
    $self->_write_worksheet_files();
}   
sub _write_worksheet_files {
    my $self = shift;
    my $dir  = $self->{_package_dir};

    _mkdir( $dir . '/xl' );
    _mkdir( $dir . '/xl/worksheets' );
    ...
}
sub _mkdir {
    my $dir = shift;

    return if -e $dir;

    my $ret = mkdir( $dir );

    if ( !$ret ) {
        croak "Couldn't create sub directory $dir: $!";
    }
}
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In Excel::Writer::XLSX::Workbook there is:

sub new {
    my $class = shift;
    my $self  = Excel::Writer::XLSX::Package::XMLwriter->new();

    ...
    $self->{_tempdir}            = undef;
    ...
}
sub _store_workbook {
    ...
    my $tempdir  = File::Temp->newdir( DIR => $self->{_tempdir} );
    ...
    $packager->_set_package_dir( $tempdir );
    $packager->_create_package();
    ...
}
sub set_tempdir {
    my $self = shift;
    my $dir  = shift;

    croak "$dir is not a valid directory" if defined $dir and not -d $
+dir;

    $self->{_tempdir} = $dir;
}
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The Workbook most likely is created with an undef _tempdir (the default). It passes a $tempdir to Packager via a call to File::Temp->newdir( DIR => $self->{_tempdir} ). E.g. for me, it created drwx------.    /tmp/4zsSPLpSBSM when given DIR=>undef. The tempdir created in this way is supposedly automatically deleted.

Then Workbook calls the packager with previous tempdir value as its _package_dir

Then Packager creates subdir 'xl', and then 'xl/worksheets'. _mkdir() will not complain if asked to re-create already existing dir (or file!, it checks -e)

I would set some debug messages as follows (in ..../perl5/Excel/Writer/XLSX/Package/Packager.pm):

sub _store_workbook {
    ...
    my $tempdir  = File::Temp->newdir( DIR => $self->{_tempdir} );
    print "tempdir='$tempdir'\n"; # debug
    if( ! -d $tempdir ){ croak "$tempdir was not created" } # debug
        my $mode = (stat($tempdir))[2];
        printf "Permissions are %04o\n", $mode & 07777;
    
    ...
    $packager->_set_package_dir( $tempdir );
    $packager->_create_package();
    ...
}
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