Many systems involve the coupled nonlinear evolution of slow and fast components, where, for example, the fast waves might be acoustic (sound) waves with a small Mach number or inertio-gravity waves with small Froude and Rossby numbers. In the past, for some such systems, an interesting property has been shown: the slow component actually evolves independently of the fast waves, in a singular limit of fast wave oscillations. Here, a fast-wave averaging framework is developed for a moist Boussinesq system with additional complexity beyond past cases, now including phase changes between water vapor and liquid water. The main question is: Do phase changes induce coupling between the slow component and fast waves? Or does the slow component evolve independently, according to moist quasi-geostrophic equations? Compared to the dry dynamics, a substantial challenge is that the method needs to be adapted to a piecewise operator with variable coefficients, due to phase changes. A formal asymptotic analysis is presented here. For purely saturated flow without phase changes, it is shown that precipitation does not induce coupling, andthe slow modes evolve independently. With phase changes present, the limiting equations show that phase boundaries could possibly induce coupling between the slow modes and fast waves. However, these possibilities were not clearly settled from theoretical considerations alone. Here, to investigate further, a suite of numerical simulations is conducted, using a sequence of small values including Fr=Ro=0.1, 0.01, and 0.001. For Fr=Ro=0.1, the influence of waves on the slow component is small, and its magnitude is roughly 0.1 to 0.4. For smaller values of Fr and Ro, the influence of waves is still somewhat small, but it does not decrease proportionally to Fr and Ro as Fr and Ro are decreased to 0.01 and 0.001. As an explanation and physical interpretation, it is shown that, while linear waves have a time average of zero, the piecewise-linear waves that arise due to phase changes actually have a nonzero time-averaged component.