Depending on the physical conditions involved, beam plasma systems may reveal new unstable regimes triggered by wave instabilities of different natures. We show through linear theory and numerical simulations the existence of an aperiodic electromagnetic instability which solely develops and controls the stability of two symmetric plasma populations counter-moving along the regular magnetic field with a relative drift, v(d), small enough to not exceed the particle thermal speed, alpha(e). Emerging at highly oblique angles this mode resembles properties of the aperiodic firehose instability driven by temperature anisotropy. The high growth rates achieved with increasing the relative drift or/and decreasing the plasma beta parameter lead to significant saturation levels of the fluctuating magnetic field power, which explain the relatively fast relaxation of electrons. For v(d) > alpha(e) this instability can coexist with the electrostatic two-stream instability, dominating the long-term dynamics of the plasma as soon as v(d) has relaxed to values smaller than the thermal speed.