Two finitely based binary operations whose union is not finitely based, and vice versa

Question

Does there exist a set $S$ and two binary operations $+$ and $*$ on that set, such that both the structures $(S;+)$ and $(S;*)$ have a finite basis of identities, but the conjoined structure $(S;+,*)$ does not have a finite basis? Also, what about vice versa? That is, both $(S;+)$ and $(S;*)$ are non-finitely based, but the structure $(S;+,*)$ is finitely based.

Answer 1

I can answer the first question.

Every finite group $\langle G; \cdot\rangle$ is finitely based (see "Identical relations in finite groups". S. Oates and B. Powell. Journal of Algebra, 1964.) We don't need the unary operation $^{-1}$ or the constant $1$ in the signature since they are encoded by $x\mapsto x^{n-1}$ and $x\mapsto x^n$ where $n=\vert G\vert$.

It seems that every set $G$ with a constant binary operation $c(x,y)=c\in G$ is described by the identity $$c(x,y)\approx c(z,w).$$

However, Roger Bryant found a finite pointed group $\langle G; \cdot, c\rangle$ that is not finitely based (see "Laws of finite pointed groups". R. Bryant. Bulletin of the LMS, 1982.) It doesn't matter if we consider $c$ as a constant or a constant binary operation.