I wonder whether from the transfer principle one can produce a satisfaction class for set-theoretic truth, which is impossible definably by Tarski's theorem.

@JamesEHanson That sounds great, if you would have any further comments about this, I'd encourage you to post an answer, even if there are only partial results. (It's fine to post as a separate answer to your other answer.)

Perhaps it is attractive to express the result as: Theorem. The following are equivalent over ZFC: (1) there is a definable global choice function, (2) any two proper class saturated models of the same theory with a definable satisfaction relation are isomorphic. In GBc, one gets the same equivalence, but the satisfaction relation needs only to exist as a class rather than be definable.

@JamesEHanson Yes, I think that works! Please post this as an answer. (But it doesn't seem to touch the main issues with the surreal numbers, and so I shall wait for further progress before accepting an answer.)

Ah, that's fine, then.

Yes, I think there is a genetic definition, but I know less about that. Perhaps Philip Ehrlich or someone else will post.

Isn't that quote about periodic functions referring not to functions defined on the Hardy fields, but rather referring to functions used to represent individual points within the Hardy field? If we are creating the Hardy field by looking at the behavior at infinity, and every individual will be positive, negative, or zero, then periodic functions like sine will be a problem.

I believe that the quotation on Wikipedia is not talking about functions on the Hardy fields, as in your question, but rather about continuous functions used to represent elements in a Hardy field. It is about the fact that elements of the Hardy field are either positive, negative, or zero, and a continuous function viewed in its behavior at infinity would not be like that.

Sorry, I don't know much that is useful about Hardy fields. Periodic functions definitely can exist in the surreals.

The point is that the same question was asked there, even if the answer is that no further progress was made.

Sorry, I don't really know what any of that means. It's not a part of how I think about the surreals.