We study the pattern formation in a two-dimensional system of superparamagnetic colloids interacting via spatially coherent induced interactions driven by an external precessing magnetic field. On the pair level, upon changing the opening angle of the external field, the interactions smoothly vary from purely repulsive (opening angle equal to zero) to purely attractive (time-averaged pair interactions at an opening angle of 90). In the experiments, we observed ordered hexagonal crystals at the repulsive end and coarsening frothlike structures for purely attractive interactions. In both of these limiting cases, the dense colloidal systems can be sufficiently accurately described by assuming pairwise additivity of the interaction potentials. However, for a range of intermediate angles, pronounced many-body depolarization effects compete with the direct induced interactions, resulting in inherently anisotropic effective interactions. Under such conditions, we observed the decay of hexagonal order with the concomitant formation of short chains and percolated networks of chains coexisting with free colloids. In order to describe and investigate these systems theoretically, we developed a coarse-grained model of a binary mixture of patchy and nonpatchy particles with the ratio of patchy and nonpatchy colloids as the order parameter. Combining genetic algorithms with Monte Carlo simulations, we optimized the model parameters and quantitatively reproduced the experimentally observed sequence of colloidal structures. The results offer new insight into the anisotropy induced by the many-body effects. At the same time, they allow for a very efficient description of the system by means of a pairwise-additive Hamiltonian, whereupon the original, one-component system features a two-component mixture of isotropic and patchy colloids.