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):
$\mathfrak{A} \equiv \mathfrak{B} \ \iff \ \exists \mathcal{U} \text{ s.t. } (\Pi_{i\in\mathcal{I}} \ \mathfrak{A})/\mathcal{U} \cong (\Pi_{i\in\mathcal{I}} \ \mathfrak{B})/\mathcal{U},$
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?
 A: Under the Continuum Hypothesis, your solution space is all nonprincipal ultrafilters. This is because under CH, the ultrapower $M^N/U$ of a mathematical structure $M$ of size at most continuum does not actually depend on the (nonprincipal) ultrafilter $U$. One can see this by using the fact that the ultrapower will be saturated, and so one can run a back-and-forth argument to achieve the isomorphism. In particular, it follows under CH that any $U$ will witness your desired isomorphism for $R^N/U\cong (R_{alg})^N/U$. 
(See Corollary 6.1.2 in Chang-Keisler's book Model Theory.) 
A similar fact holds for larger cardinals and larger structures under GCH, but here, one needs an additional assumption on the ultrafilter. Namely, Theorem 6.1.9 in Chang-Keisler asserts that if $2^\alpha=\alpha^+$ and $A$ and $B$ are two structures of size at most $\alpha^+$, then they are elementarily equivalent if and only if $\Pi_DA\cong\Pi_D B$ for any $\alpha^+$-good incomplete ultrafilter $D$ on $\alpha$. The proof uses the same saturation idea, and this establishes the Keisler-Shelah theorem in the case that GCH holds. 
Chang-Keisler states (page 393-394) that it is open whether the assertion of Theorem 6.1.9 stated above holds under $\neg CH$. 
A: As you might expect, things are consistently much more interesting if $CH$ fails. This has been explored by Shelah in a fascinating series of papers "Vive la difference I - III". For example, it is consistent that there is a nonprincipal ultrafilter $\mathcal{U}$ on $\omega$ such that if $(R_{n})$ and $(S_{n})$ are sequences of discrete rank 1 valuation rings having countable residue fields, then any isomorphism $\varphi: \prod_{\mathcal{U}}R_{n} \to \prod_{\mathcal{U}}S_{n}$ is an ultraproduct of isomorphisms $f_{n}: R_{n} \to S_{n}$. In particular, $\mathcal{U}$-almost all $R_{n}$ are isomorphic to the corresponding $S_{n}$ and so the Ax-Kochen isomorphism theorem doesn't hold with respect to $\mathcal{U}$.
If you are only interested in ultraproducts of fixed structures $A$, $B$, then I should mention that it is also consistent that there exists an ultrafilter $\mathcal{A}$ on $\omega$ such that if $A$ and $B$ are countable structures which satisfy the strong independence property, then the corresponding $\mathcal{A}$-ultraproducts are isomorphic iff $A \cong B$.
