lo.logic - Ultrafilters arising from Keisler-Shelah ultrapower characterisation of elementary equivalence

In model theory, two structures $mathfrak{A}, mathfrak{B}$ of identical signature $Sigma$ are said to be elementarily equivalent ($mathfrak{A} equiv mathfrak{B}$) if they satisfy exactly the same first-order sentences w.r.t. $Sigma$. An astounding theorem giving an algebraic characterisation of this notion is the so-called Keisler-Shelah isomorphism theorem, proved originally by Keisler (assuming GCH) and then by Shelah (avoiding GCH), which we state in its modern strengthening (saying that only a single ultrafilter is needed):

$mathfrak{A} equiv mathfrak{B} iff exists mathcal{U} text{ s.t. } (Pi_{iinmathcal{I}} mathfrak{A})/mathcal{U} cong (Pi_{iinmathcal{I}} mathfrak{B})/mathcal{U},$

where $mathcal{U}$ is a non-principal ultrafilter on, say, $mathcal{I} = mathbb{N}$. That is, two structures are elementarily equivalent iff they have isomorphic ultrapowers.

My question is the following (admittedly rather vague): Does anyone know of constructions in which an ultrafilter is chosen by an appeal to this characterisation and then used for other means? An example of what I have in mind would be something like this (using the fact that any two real closed fields are elementarily equivalent w.r.t. the language of ordered rings): In order to perform some construction $C$ I ``choose'' a non-principal ultrafilter $mathcal{U}$ on $mathbb{N}$ by specifying it as a witness to the following isomorphism induced by Keisler-Shelah:

$mathbb{R}^mathbb{N}/mathcal{U} cong mathbb{R}_{alg}^mathbb{N}/mathcal{U},$

where $mathbb{R}_{alg}$ is the field of real algebraic numbers. So the construction $C$ should be dependent upon the fact that $mathcal{U}$ is a non-principal ultrafilter bearing witness to the Keisler-Shelah isomorphism between some ultrapower of the reals and the algebraic reals, resp.

Also, a follow-up question: Let's say I'd like to ``solve'' the above isomorphism for $mathcal{U}$. Are there interesting things in general known about the solution space, e.g., the set of all non-principal ultrafilters bearing witness to the Keisler-Shelah isomorphism for two fixed elementarily equivalent structures such as $mathbb{R}$ and $mathbb{R}_{alg}$? What machinery is useful in investigating this?