Given two covering maps $p\colon \tilde{X} \to X$ and $q\colon \tilde{Y} \to Y$, we can form the covering map $p\times q \colon \tilde{X} \times \tilde{Y} \to X\times Y$. By covering space theory, we know that covering maps of $X\times Y$ correspond to subgroups of $\pi_1(X) \times \pi_1(Y)$. So not all covering maps of $X\times Y$ should be in this product form (since, e.g., we can consider diagonal subgroups).
However, is it true that all covering spaces (the total spaces) of $X\times Y$ are homeomorphic to $\tilde{X}\times \tilde{Y}$ where $\tilde{X}$ covers $X$ and $\tilde{Y}$ covers $Y$? (This is true for the torus for example.)