Particle-in-cell (PIC) methods are the standard for simulating non-equilibrium plasmas due to the challenges in directly discretizing the Boltzmann equation, which suffers from the curse of dimensionality. Instead, PIC methods evolve samples (called “particles”) of an initial, prescribed distribution function subject to collisions and Lorentz forces from self-consistently computed electromagnetic fields. However, as a Monte Carlo method, PIC suffers from slow error convergence (∼ 1/√N), and simulations of far-from-equilibrium plasmas may necessitate a greater number of particles than is computationally practical. This limitation has inspired work toward methods that imbue particles with some information about the local value of the distribution in order to improve the error convergence, akin to variance reduction techniques. Extension of these methods to collisional regimes has nonetheless proven challenging and often results in numerical instability, requiring much effort to eliminate [1, 2, 3]. Recent work has explored a flow-based scheme for evolving the distribution along phase space trajectories, but has not yet been applied beyond simple collision kernels [4]. This work presents a method for retaining local distribution information on PIC particles while using the Monte Carlo collisions method [5]. The value of the distribution function is stored directly on particle positions in phase space. Since most quantities of interest in PIC simulations are derived from moments of the distribution, this allows for the use of variance reduction techniques. Such variance reduction techniques involve the auxiliary simulation of a cheap low-fidelity distribution to act as a reference distribution. Since most quantities of interest in PIC simulations are moments of the distribution or derived therefrom, the convergence acceleration afforded by variance reduction ultimately promises to reduce the particle count compared to similarly resolved PIC simulations. Throughout, considerations are made to ease programming into existing PIC codes. Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525.