Under some additional set-theoretic assumptions, we construct a totally disconnected, non-metrizable compact space $K$ such that the Banach space $C(K)$ admits uncountable biorthogonal systems, but does not admit such systems consisting solely of atomic measures. It follows from results of Todorčević that such a construction cannot be executed in ZFC.

The space $K$ is a version of a split Cantor set, obtained by one-by-one splitting some $\omega_1$ points of the Cantor set into new Cantor sets.