  As proven in a celebrated theorem due to Vaisman, pure locally conformally
K\"ahler metrics do not exist on compact K\"ahler manifolds. In a previous
paper, we extended this result to the singular setting, more precisely to
K\"ahler spaces which are locally irreducible. Without the additional
assumption of local irreducibility, there are counterexamples for which
Vaisman's theorem does not hold. In this article, we give a much broader
sufficient condition under which Vaisman's theorem still holds for compact
K\"ahler spaces which are locally reducible.
