What do you call a surgery without anesthesia? I think there are several phrases you can use to refer to "surgery that happens without anesthesia", but I don't remember the idiomatic expressions that are usually used to refer to it.
You would normally only need this in an historical context. You should just say "without anaesthetic" Consider https://www.ncbi.nlm.nih.gov/books/NBK481552/ which is a long article that doesn't use any specialist term, yet uses other medical language (lithotomy is the removal of a bladder stone)
The mere thought of having a leg amputated without any kind of anaesthetic, let alone submitting to a protracted lithotomy, [...] is liable to make the modern reader flinch in sympathy.
I don't think there is an idiomatic way of saying this, as such a thing is very rare. Surgery implies that cutting the body in some way is involved (except for dental surgery), which one would always have at least a local anesthesia for. It appears that there may be a medical term for using hypnosis instead of anesthesia, but this is is not really idiomatic, as most people don't know of it.
It also may be that you are thinking that having anesthesia means you are unconscious, which is not necessarily the case.