Suppose $\mathscr{V}$ is a pseudovariety in a countable language $\Sigma$. Say that a minimal description of $\mathscr{V}$ (in the sense of Eilenberg/Schutzenberger rather than Reiterman) is a pair $(E,F)$ of sets of $\Sigma$-equations such that
$\mathscr{V}$ is the set of finite $\Sigma$-algebras which satisfy all equations in $E$ and all-but-finitely-many equations in $F$; and
if $E'\subseteq E, F'\subseteq F$ are such that $(E', F')$ satisfies the previous bulletpoint, then $E'=E$ and $\vert F\setminus F'\vert<\infty$.
In a previous question, I asked about the possible number of minimal descriptions up to an appropriate notion of equivalence. Following comments by Benjamin Steinberg, I want to take a step back and ask a coarser question:
Which pseudovarieties have any minimal descriptions in the first place?
I'm also interested in the same question relativized to a variety $\mathsf{V}$: given a variety $\mathsf{V}$ and a pseudovariety $\mathscr{V}\subseteq \mathsf{V}$, say that a $\mathsf{V}$-minimal description of $\mathscr{V}$ is a pair $(E,F)$ satisfying the conditions above with "finite $\Sigma$-algebras" replaced by "finite $\Sigma$-algebras in $\mathsf{V}$." That said, I am primarily interested in the "pure" version of the question above.
