Quantum interference phenomena are widely viewed as posing a challenge to the classical worldview. Feynman even went so far as to proclaim that they are the only mystery and the basic peculiarity of quantum mechanics. We challenge this claim by describing an alternative to quantum theory, a statistical theory of a classical discrete field (the "toy field theory") that reproduces the relevant phenomenology of quantum interference while respecting the classical worldview. It also reproduces a number of related interference experiments, such as the Elitzur-Vaidman bomb tester, Wheeler's delayed-choice experiment, and the quantum eraser experiment.