We present a model study of a multi-component system that can form low-symmetry ordered phases, even though all pair interactions between the constituent particles are spherically symmetric. Using Monte Carlo simulations and a mean-field model we investigate the thermodynamic, structural, and kinetic aspects of the formation of stripe phases for a simple, multi-component lattice model. This lattice model was chosen to represent a mixture of spherically symmetric DNA-coated colloids with several species of DNA linkers. We predict that the optimal strategy to design a specific low-symmetry phase is the one which not only provides the weakest strength of competitive binding, but also leads to the 'staged' ordering of nanoparticle superstructures.