It follows from the definition of a completely regular space that such spaces have a base consisting of co-zero sets, that is, sets whose complement is the zero set of some real-valued, continuous function.
On the other hand, it is not hard to show that regularly open sets also form a base. (A set is regularly open if it is equal to the interior of its closure.) So my question is
Does every compact Hausdorff space have a base which consists of sets which are simultaneously regularly open and co-zero?
Of course, the question is meaningful for spaces which are not metrisable (or more generally, not perfectly $\kappa$-normal).
I am not really interested in special cases, but in full generality.