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.