This article is concerned with the existence and uniqueness of solutions of the Cauchy problem in the periodic setting for a regularized Benjamin-Ono type system (rBO) by using semigroup theory, Fourier analysis and Banach's fixed point theorem.This system was recently derived by Muñoz [12] as a weakly dispersive model for the propagation of small amplitude internal waves at the interface of two immiscible fluids with constant densities.We also conduct some numerical experiments to analyze the error and convergence in time and space of a fully discrete Fourier spectral scheme, for approximating the solutions of the initial value problem associated to the rBO system.