Semi-proximal spaces and normality

We consider the relationship between normality and semi-proximality. We give a consistent example of a first countable locally compact Dowker space that is not semi-proximal, and two ZFC examples of semi-proximal non-normal spaces. This answers a question of Nyikos. One of the examples is a subspace of $(\omega+1) \times \omega_1$. In contrast, we show that every normal subspace of a finite power of $\omega_1$ is semi-proximal.