Let $X$ be a compact Kähler Calabi-Yau manifold, and $\alpha\in H^{1,1}_{\mathbb{R}}(X)$ be a class which is nef and big, but not Kähler. Then $\alpha$ can be represented by a smooth $(1,1)$ form $\omega$ which is pointwise nonnegative and which is Kähler outside a proper analytic subvariety $E\subset X$.