  A central problem in quantum information is determining quantum-classical
boundaries. A useful notion of classicality is provided by the quasiprobability
formulation of quantum theory. In this framework, a state is called classical
if it is represented by a quasiprobability distribution that is positive, and
thus a probability distribution. In recent years, the Kirkwood-Dirac (KD)
distributions have gained much interest due to their numerous applications in
modern quantum-information research. A particular advantage of the KD
distributions is that they can be defined with respect to arbitrary
observables. Here, we show that if two observables are picked at random, the
set of classical states of the resulting KD distribution is a simple polytope
of minimal size. When the Hilbert space is of dimension $d$, this polytope is
of dimension $2d-1$ and has $2d$ known vertices. Our result implies,
$\textit{e.g.}$, that almost all KD distributions have resource theories in
which the free states form a small and simple set.
