Volume-averaged flow equations model fluid systems with two or more interpenetrating phases. Such models are used in various engineering and science applications, including chemical and geophysical flows. Each fluid obeys its own set of Navier-Stokes equations, and the interphase coupling occurs via mass conservation, drag forces, and a common pressure shared by all phases. Therefore, designing decoupling schemes to avoid costly monolithic solvers is a complex, yet very relevant task. In particular, it requires treating the pressure explicitly in a stable way. To accomplish that, we present a novel incremental pressure-correction method built upon the fact that the mean (volume-averaged) flow field is incompressible, even though each individual phase may have a non-solenoidal velocity. To completely and stably decouple the phase equations, the drag is made implicit-explicit (IMEX). Furthermore, by treating all nonlinear terms in a similar IMEX fashion, the new method completely eliminates the need for Newton or Picard iterations. At each time step, only linear advection-diffusion-reaction and Poisson subproblems need to be solved as building blocks for the multi-phase system. The resulting time-stepping scheme is unconditionally stable, that is, no CFL conditions arise. The stability and first-order temporal accuracy of the method are illustrated via two-phase numerical examples using finite elements for the spatial discretisation.