Let $\mathcal{P}$ be a set of $n$ polygons in $\mathbb{R}^3$, each of constant complexity and with pairwise disjoint interiors. We propose a rounding algorithm that maps $\mathcal{P}$ to a simplicial complex $\mathcal{Q}$ whose vertices have integer coordinates. Every face of $\mathcal{P}$ is mapped to a set of faces (or edges or vertices) of $\mathcal{Q}$ and the mapping from $\mathcal{P}$ to $\mathcal{Q}$ can be done through a continuous motion of the faces such that (i) the $L_\infty$ Hausdorff distance between a face and its image during the motion is at most 3/2 and (ii) if two points become equal during the motion, they remain equal through the rest of the motion. In the worst case the size of $\mathcal{Q}$ is $O(n^{13})$ and the time complexity of the algorithm is $O(n^{15})$ but, under reasonable assumptions, these complexities decrease to $O(n^{4}\sqrt{n})$ and $O(n^{5})$. Furthermore, these complexities are likely not tight and we expect, in practice on non-pathological data, $O(n\sqrt{n})$ space and time complexities.