  The Kirkwood-Dirac (KD) quasiprobability distribution can describe any
quantum state with respect to the eigenbases of two observables $A$ and $B$. KD
distributions behave similarly to classical joint probability distributions but
can assume negative and nonreal values. In recent years, KD distributions have
proven instrumental in mapping out nonclassical phenomena and quantum
advantages. These quantum features have been connected to nonpositive entries
of KD distributions. Consequently, it is important to understand the geometry
of the KD-positive and -nonpositive states. Until now, there has been no
thorough analysis of the KD positivity of mixed states. Here, we characterize
how the full convex set of states with positive KD distributions depends on the
eigenbases of $A$ and $B$. In particular, we identify three regimes where
convex combinations of the eigenprojectors of $A$ and $B$ constitute the only
KD-positive states: $(i)$ any system in dimension $2$; $(ii)$ an open and dense
set of bases in dimension $3$; and $(iii)$ the discrete-Fourier-transform bases
in prime dimension. Finally, we investigate if there can exist mixed
KD-positive states that cannot be written as convex combinations of pure
KD-positive states. We show that for some choices of observables $A$ and $B$
this phenomenon does indeed occur. We explicitly construct such states for a
spin-$1$ system.
