I don't know that this "presentation" is too much more familiar, but the covering space you're interested in is diffeomorphic to $S^2\times S^2/\sim$ where $(x,y) \sim \pm(x,y)$. The projection map sends $[(x,y)]$ to $([x],[y])$.
Now, $S^2\times S^2/\sim$ is diffeomorphic to another relatively well known space: the Grassmanian of unoriented $2$-planes in $\mathbb{R}^4$.
This space is not homotopy equivalent to any of the other covering spaces of $\mathbb{R}P^2\times\mathbb{R}P^2$. Its fundamental group is $\mathbb{Z}/2\mathbb{Z}$, so it could only potentially be homotopy equivalent to $\mathbb{R}P^2\times S^2$ or $S^2\times\mathbb{R}P^2$. On the other hand, both of these second two spaces are nonorientable, while $S^2\times S^2/\sim$ is orientable.