Stable vortex lattices are basic dynamical patterns which have been widely demonstrated in many physical systems including superconductor physics, Bose-Einstein condensates, hydrodynamics and optics. Vortex-antivortex (VAV) ensembles can be produced too, self-organizing into the respective polar lattices. However, these structures are in general highly unstable due to the strong VAV attraction. Here, we demonstrate that multiple optical VAV clusters nested in the propagating coherent field can crystallize into new patterns which preserve their lattice structure in the course of the linear propagation over distance up to several Rayleigh lengths. To explain this phenomenon, we present a model for effective interactions between the vortices and antivortices located at different sites of the lattice. We conclude that the observed VAV crystallization is a consequence of the globally balanced VAV couplings. As the crystallization does not require the presence of nonlinearities and takes place in the free space, it may find promising applications to high-capacity optical communications and multiparticle manipulations by means of optical fields. Our findings also suggest new possibilities for constructing VAV complexes by means of the orbit-orbit coupling mechanism, which differs greatly from the extensively studied spin-orbit couplings.