Not yet reviewed by Pith; the record is open.

The Numerical Flow Iteration (NuFI) method has recently been proposed as a memory-slim while accurate in phase-space method for the electro-static Vlasov--Poisson system. It stores the temporal evolution of the electric field, instead of the distribution functions, and reconstructs the solution in each time step by following the characteristics backwards in time and reconstructing the solution from the initial distribution. NuFI has been shown to be more accurate than other state-of-the-art electro-static Vlasov solvers given the same amount of degrees of freedom. In this paper, we build on the Hamiltonian structure of the full Vlasov--Maxwell system to extend NuFI to handle electro-magnetic kinetic plasma dynamics. We show that the structure-preserving properties of the NuFI time-stepping are preserved when extending to the electro-magnetic case. Furthermore we discuss how NuFI can be incorporated into existing Semi-Lagrangian codes as an efficient while accurate subcycling technique.