Can there be a computational characterization of HOD?

This is a somewhat open-ended question, so let me motivate it a little.

One of the first results in the study of Ordinal Turing Machines includes the following computational characterization of $$L$$ by Koepke:

Theorem. a set of ordinals is in $$L$$ if and only if it is computable by an Ordinal Turing Machine with finitely many parameters.

At the same time, $$\mathsf{HOD}$$ has a "definable power set" characterization: in the construction on $$L$$, if we replace the first-order definable power set operator with second-order definable power set, then the resulting structure is $$\mathsf{HOD}$$. This is proved by Myhill and Scott, and is also nicely written up in this MO post.

So a natural question is this: can there be a computational notion $$C$$ such that something analogous to Koepke's theorem is provable for $$\mathsf{HOD}$$? By this I have in mind something to the effect of "a set (of ordinals) is in $$\mathsf{HOD}$$ if and only if it is $$C$$-computable."

A somewhat trivial candidate would be just oracle-OTM computation. This uses the observation that $$\mathsf{HOD}$$ is always coextensive with $$L[A]$$ for some class of ordinals $$A$$, and we can just relativize Koepke's theorem to $$A$$. But can we do more?

• A question by Dmytro Taranovsky might be relevant. (Unfortunately, it does not receive any answer.) – Hanul Jeon Feb 1 at 9:41
• At some point you gotta ask yourself what counts as a computation. If you need a 2nd-order oracle mechanism, then is it really a computation? – Asaf Karagila Feb 1 at 12:28