  We prove a semi-global gauge-invariant estimate for the solutions of the
characteristic initial value problem associated with the coupled
Einstein-Yang-Mills equations. In particular, we prove the existence of
\textit{a} future development of regular initial data on a pair of incoming and
outgoing null hypersurfaces emanating from a spacelike topological $2$-sphere.
This marks the first study of the characteristic initial value problem of
Einstein's equations with a non-linear source.
