In Keisler's Infinitary Logic Book, he notes at one point that if we take a real closed field of cardinality $\kappa$, the ordered set is countably strongly homogeneous, in the sense we can find an automorphism taking any increasing finite $\overline{a}$ to $\overline{b}$.
I do not entirely understand why this is sufficient. This is not a consequence of the order being a dense linear order, as there are DLOs which are do not have this property, such as by taking the order type of the reals and end-extending by the order type of the rationals.
It is also clear that there are real closed fields that are not countably strongly homogeneous as fields, such as $\mathbb{R}$, we cannot just take the reduct of a stronger automorphism. The theory itself is also unstable, and so does not guarantee even homogeneous models in every cardinality.
What additional property of real closed fields ensures that the order type has this property?
Apologies for any errors, I have a migraine and both bright screens and thinking are rather difficult.
The point is that if $a<a'$ and $b<b'$ in a real closed field, then the linear map $$x\mapsto \frac{b'-b}{a'-a}(x-a)+b$$ is an order-preserving bijection between the intervals $[a,a']$ and $[b,b']$. Similarly, for any $c$ and $d$, the map $$x\mapsto x+(d-c)$$ is an order-preserving bijection between the intervals $(-\infty,c]$ and $(-\infty,d]$, and between the intervals $[c,\infty)$ and $[d,\infty)$. Now you can cobble together functions of these forms into a piecewise linear map which is an isomorphism of ordered sets and maps any finite increasing tuple to any other of the same length.