Kirkwood-Dirac (KD) quasiprobability distributions are increasingly used across quantum information science, yet their computational significance remains unclear. We classify unitary dynamics that preserve KD positivity and connect this structure to classical simulation. In contrast with the discrete Wigner setting, we show that KD positivity preservation, stochastic evolution of quasiprobabilities, and preservation of total nonpositivity do not coincide. We identify three classes of positivity-preserving unitaries: type I gates, which are exactly the KD-stochastic unitaries; type II gates, which are nonstochastic yet preserve total nonpositivity by permuting and conjugating KD entries; and, for Fourier-conjugate bases in dimension d = pq, type III gates, which preserve KD positivity but can change the total nonpositivity of nonreal distributions. The classification is complete for Haar-random bases and for Fourier-conjugate dimensions d = pk and d = pq, with p, q distinct primes. Adapting the sampling algorithm of Pashayan et al. [Phys. Rev. Lett. 115, 070501 (2015)], we simulate all of these positivity-preserving circuits efficiently on KD-positive inputs; type III gates, however, create a sharp distinction between real inputs, which remain efficiently simulable, and nonreal inputs, whose sampling overhead can grow exponentially. Consequently, for d = pq no resource theory can both treat the KD total nonpositivity as a monotone and admit every efficiently simulable positivity-preserving unitary as a free operation.
