Electrospheres are environments with the same origin as pulsars; a highly magnetized rotating neutron star. In pulsars, a cascade of electron-positron pair creation enriches the plasma. The plasma surrounding an electrosphere consists only of particles that have escaped from the neutron star's surface. Electrospheres with a magnetic axis aligned with the rotation axis have been well described for decades. Models of electrospheres with an oblique magnetic axis relative to the rotation axis have resisted most theoretical investigations. Some electrospheres and pulsars have been simulated using particle-in-cell codes, but the numerical constraints did not allow the use of realistic neutron star parameters. We aimed to develop a numerical simulation code optimized for understanding the physics of electrospheres and pulsars, with realistic neutron star parameters. As a first step, presented in this paper, we focused on the simulation of oblique electrospheres with realistic physical parameters. A specific code was developed for the computation of stationary solutions. The resolution of Maxwell's equations was based on spectral methods. Particle motions included their finite inertia. No hypothesis was made in relation to the force-free behavior of the electrospheric plasma. The numerical code is called Pulsar ARoMa (pulsar asymmetric rotating magnetosphere). Various numerical simulations were conducted using realistic neutron star parameters. We find that oblique electrospheres possess the same global structure as aligned force-free electrospheres, with two domes of electrons and a torus of positively charged particles. The domes are not centered on the magnetic axis; nor are they symmetric. Yet, the solutions do not exhibit a force-free behavior. The simulations performed with the Pulsar ARoMa code require modest resources and little computing time. This code will be upgraded for more ambitious investigations into pulsar physics.