Monkeying around further I found that adding this to the end of the example:

$data->{numref} = \$data->{nums}->[0];
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Produces this after the first pass:

$VAR1 = {
          'nums' => [
                      '-=ONE=-',
                     ...],
          'numref' => \'-=-=ONE=-=-',
          ...
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The link is lost and the leaf is visited twice.

Repeating the "-=\U$_=-" line, gives

'nums' => ['-=-=ONE=-=-', ...
'numref' => \'-=-=-=ONE=-=-=-', ...
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Not likely to be a problem, but worth the note.
Update:possible fix:

$refs->{ref($_) ? $_: \$_} = undef;