Discussion on answer by Lukas Heger: If $(\mathbb R,<,+,\boxdot)$ is an ordered division ring, is it automatically a field?

Imported from a comment discussion on https://math.stackexchange.com/questions/5070122/if-mathbb-r-boxdot-is-an-ordered-division-ring-is-it-automatically-a/5070160#5070160