Finite algebras with non-computable morphisms

We construct a variety $\mathbf V$ of partial algebgras with a finite basis of Kleene identities and a computable sequence $(\mathcal A_n \mid n< \omega)$ of finite algebras in $\mathbf V$ with a non-computable set $\{n \mid \mathcal A_n\ \text{is simple in}\ \mathbf V\}$, where the property ‘simple’ is considered with respect to epimorphisms.