It is a well-known fact from introductory real analysis that $\mathbb{R}$ is the unique Dedekind-complete ordered field, up to isomorphism. Hence, we are in a sense justified in an abstract definition of the reals as being a complete ordered field. Often the reals are explicitly constructed via Dedekind cuts or by constructing the completion of $\mathbb{Q}$ via Cauchy sequences, and it can be shown that these two constructions are equivalent.
I was wondering whether there are any other examples of ordered fields that are non-trivially seen to be isomorphic to the real numbers by the uniqueness of $\mathbb{R}$. More specifically, I want to know whether there are examples of complete ordered fields that are not obviously isomorphic to $\mathbb{R}$, but are seen to be isomorphic via the stated fact.