  As a variant of the equal entropy cover problem, we ask whether all
multidimensional sofic shifts with countably many configurations have SFT
covers with countably many configurations. We answer this question in the
negative by presenting explicit counterexamples. We formulate necessary
conditions for a vertically periodic shift space to have a countable SFT cover,
and prove that they are sufficient in a natural (but quite restricted) subclass
of shift spaces.
