To understand how quantum mechanics works, it is useful to imagine alternative “foil” theories that work differently. Modal quantum theory is a discrete toy model that is similar in structure to ordinary quantum theory, but based on a finite field instead of complex amplitudes. The interpretation of this theory involves only the “modal” concepts of possibility and impossibility rather than quantitative probabilities. Despite its very simple structure, our toy model nevertheless includes many of the key features of actual quantum physics: interference, complementarity, entanglement, teleportation, the impossibility of cloning, pseudo-telepathy games, nonclassical computation, and more.