Let $M$ be a model and $N$ be an elementary submodel of $M$. We say $\bar{a} \in M ^{n}$ is countably determined over $N$ if $\mathrm{tp}^M(\bar{a}/N) := \{A \in \mathrm{Def}^n(M,N) \mid \bar{a} \in A \}$ is countably generated where $\mathrm{Def}^n(M,N)$ is the set of all definable sets in $M^n$ with parameter in $N$. We say $M$ is countably determined over $N$ if every $\bar{a} \in M ^{< \omega}$ is countably determined.
Suppose that every $a \in M $ is countably determined over $N$. Then, is $M$ is countably determined over $N$?
If not, how about the case $M$ and $N$ are real closed ordered fields?