Someone else here brought up the surreal numbers and my intuition says that it's right to do that. The various arithmetic proofs thrown around here don't explicitly make use of completeness. As such they should be correct proofs in the surreal numbers as well. But they basically are not. Here is a blog post about it:

https://thatsmaths.com/2019/01/10/really-0-999999-is-equal-t...

I don't quite know how to formalize it, but I'm pretty certain that if these proofs logically worked (in the "theory of proofs sense"), then they should work in the surreals as well.

Anyway it's just intuition. My main point in this thread is that I don't really accept the proofs of this that don't use completeness as a step. Though I do suspect that proofs not making use of it are actually incorrect proofs in their own right. If I were curious enough I'd think back about formal proofs and models and all that jazz, but I probably already have spent more time in this thread than I should. :)

edit: The more I think about it I feel like someone actually explained to me this (i.e. why this proof is wrong using surreals as reasoning) a long time ago and I'm just remembering echos of it in my mind. Wish I could remember something more useful...or that I were a logician...