Under the Continuum Hypothesis, your solution space is all nonprincipal ultrafilters. This is because under CH, the ultrapower $M^I/U$ 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 $R^N/U\cong (R_{alg})^N/U$.

A similar fact holds for larger cardinals and larger structures under GCH.

I'm not sure what happens under $\neg$CH. I have a vague recollection that the proof of the Keisler-Shelah theorem was quite flexible with the choice of $U$, but perhaps the model theorists here can say more precisely.

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 $\neg CH$.

Under the Continuum Hypothesis, your solution space is all nonprincipal ultrafilters. This is because under CH, the ultrapower $M^I/U$ 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 $R^N/U\cong (R_{alg})^N/U$.

I'm not sure what happens under $\neg$CH. I have a vague recollection that the proof of the Keisler-Shelah theorem was quite flexible with the choice of $U$, but perhaps the model theorists here can say more precisely.

Under the Continuum Hypothesis, your solution space is all nonprincipal ultrafilters. This is because under CH, the ultrapower $M^I/U$ 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 $R^N/U\cong (R_{alg})^N/U$.

A similar fact holds for larger cardinals and larger structures under GCH.

I'm not sure what happens under $\neg$CH. I have a vague recollection that the proof of the Keisler-Shelah theorem was quite flexible with the choice of $U$, but perhaps the model theorists here can say more precisely.