  We construct a toy model for the harmonic oscillator that is neither
classical nor quantum. The model features a discrete energy spectrum, a ground
state with sharp position and momentum, an eigenstate with non-positive Wigner
function as well as a state that has tunneling properties. The underlying
formalism exploits that the Wigner-Weyl approach to quantum theory and the
Hamilton formalism in classical theory can be formulated in the same
operational language, which we then use to construct generalized theories with
well-defined phase space. The toy model demonstrates that operational theories
are a viable alternative to operator-based approaches for building physical
theories.
