The call stack depth as a whole is not usually what interests me but more often the depth of recursion, which more often corresponds with the depth of something in the input.

In that case it's a simple matter of incrementing a depth counter on entry and decrementing it just before the return, e.g.

sub gettag {
    my $self = shift;
    ($self -> { THIS }{ DEPTH } ||= 0)++;
    my $tagref = { SUBTAG => [] };
    # ...
    while ( !/<\// ) {
        push @{$tagref -> { SUBTAG }},
          $self -> gettag(); # recursive call
    }
    # ...
    $self -> { THIS }{ DEPTH }--;
    return $tagref;
}
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