We study a coupled electron-ion plasma governed by the Boltzmann-Poisson equations in a regime where intense electric and magnetic fields dominate. Although the configuration under study remains generic, it is strongly inspired by the physical conditions encountered in Hall thrusters. A scaling analysis reveals that electrons undergo complex dynamics: they combine an ExB drift with a rapid cyclotron gyration around the magnetic field lines, which leads to their trapping along these lines. In contrast, the ions remain weakly magnetized but are strongly accelerated by the electric field. This multiscale behavior introduces significant numerical stiffness in the governing partial differential equations, mainly (though not exclusively) due to the very fast electron gyromotion. To address this difficulty, we propose a global modeling and numerical strategy for the electrons based on a fluid reduction. First, at the kinetic level, a Hilbert expansion separates the distribution into a leading-order gyroaveraged equilibrium (i.e., the averaged dynamics of the trajectories once the fast cyclotron oscillations have been filtered out) and higher-order oscillatory corrections associated with the fast cyclotron motion. This structure is then exploited to derive a moment-based fluid model using a tailored polynomial basis that distinguishes gyroaveraged and oscillatory contributions. The resulting “gyromoment” model captures the anisotropic features of the strongly magnetized regime while filtering out the fastest time scales associated with numerical stiffness. The ions are also described using a fluid model, leading to a fully fluid electron-ion description. In contrast to the electrons, the ion fluid model follows the standard Euler-like formulation, as their weak magnetization does not require "gyro-reduction". Finally, we present numerical experiments illustrating the multiscale plasma dynamics and the relevance of the proposed reduced model in regimes where direct kinetic simulations are prohibitively stiff.