Non-basic pseudo-elementary class

Question

Is there a pseudo-elementary class, which is not an elementary class itself, of structures which is not basic pseudo-elementary, that is, is not the reduct of a basic elementary class?

Answer 1

Note that if $\mathbb{K}$ is a basic pseudoelementary class in a finite language, then $Th(\mathbb{K})$ is recursively enumerable: $Th(\mathbb{K})$ consists exactly of those sentences in that language which are provable from $\varphi$ (where $\mathbb{K}$ is the set of reducts of models of $\{\varphi\}$). So any non-recursively enumerable, deductively-closed set of sentences in a finite language provides an elementary class which is not basic pseudoelementary.

For a natural example, we can take $\mathbb{K}$ to be the set of models of true arithmetic.