(originally asked at M.SE: https://math.stackexchange.com/questions/4094361/first-time-the-reals-were-axiomatized-as-the-unique-complete-ordered-field)

I'm looking for historical references on the history of the axiomatization of real numbers. I found out that Tarski had an axiomatization of the reals as in https://en.wikipedia.org/wiki/Tarski%27s_axiomatization_of_the_reals. However, the usual approach to the reals is as the "unique complete ordered field", which is somewhat different.

Who first described the reals as the "unique complete ordered field"?

Any other axiomatizations which have appeared throughout history are also welcome.