The Hewitt realcompactification of products

For a completely regular Hausdorff space $X$, $\upsilon X$ denotes the Hewitt realcompactification of $X$. Given a topological property $\mathcal{P}$ of spaces, our interest is in characterizing the class $\mathcal{R}(\mathcal{P})$ of all spaces $X$ such that $\upsilon (X \times Y) = \upsilon X \times \upsilon Y$ holds for each $\mathcal{P}$-space $Y$. In the present paper, we obtain such characterizations in the case that $\mathcal{P}$ is locally compact and in the case that $\mathcal{P}$ is metrizable.