A novel splitting scheme to solve parametric multi-convex programs is presented. It consists of a fixed number of proximal alternating minimisations and a dual update per time step, which makes it attractive in a real-time Nonlinear Model Predictive Control (NMPC) framework and for distributed computing environments. Assuming that the parametric program is semi-algebraic and that its critical points are strongly regular, a contraction estimate is derived and it is proven that the sub-optimality error remains stable under some mild assumptions. Efficacy of the method is demonstrated by solving a bilinear NMPC problem to control a DC motor. In particular, the effect of the sampling period on the optimality tracking error is analysed for a fixed computational power.