In this article, we justify the use of “geometric optics” like methods for a large of semilinear systems of field equations, that remain invariant under gauge and Lorentz transforms. In a first time, we consider a model system, coupling a gauge field (Yang–Mills equation) with a scalar field (wave equation) and a spinor field (Dirac equation), and build families of approximate solutions in the shape of single phase expansions, rapidly oscillating at high frequency. Amplitude of such oscillations is maximal with regard to their frequency parameter. Expansions are given to any order. In a second time, we give exact solutions that remain asymptotic to previous oscillatory expensions. Such solutions may be uniformly bounded only in $H^{1/2},H^{1/2}$ and $L^2$ — respectively for gauge, scalar and spinor fields. We give a stability result in the case where oscillatory solutions are obtained as high frequency perturbations of a given smooth solution of the system. It shows that the system can keep a stable nonlinear behaviour, even for very low regularity fields, when nonlinear terms usually lead to destructive interactions.