There are counterexamples to order isomorphisms of ordered fields being field isomorphisms, see Is the multiplicative structure of a totally ordered field unique?. However, Wikipedia suggests that for real closed fields order isomorphism implies isomorphism: "If the continuum hypothesis holds, all real closed fields with cardinality the continuum and having the $η_1$ property are order isomorphic. This unique field $F$ can be defined by means of an ultrapower... This is the most commonly used hyperreal number field in non-standard analysis, and its uniqueness is equivalent to the continuum hypothesis."
How exactly does one recover operations from order if $F$ is real closed?