TITLE: Non-isomorphic structures of the same cardinality which are elementary equivalent
QUESTION [15 upvotes]: We usually give examples (es. ultrapowers) of elementary equivalent structures which are not isomorphic because they have different cardinalities?
Can someone help me to find an example of two such structures which have the same cardinality?
Can some constructions with ultra products help? (E.g. involving hypotheses about their cardinality, taking regular ultrafilters and similar).
Thank you in advance.

REPLY [3 votes]: The countable ultrapower of $\mathbb R$ actually does not increase the cardinality so $\mathbb{R}^{\mathbb N}/\mathcal{F}$ where $\mathcal{F}$ is a free ultrafilter on $\mathbb N$ is an example of an elementary extension that does not increase cardinality.