  We study the local well-posedness of the Einstein-Yang-Mills equations in
constant mean extrinsic curvature spatial harmonic generalized Coulomb gauge
(CMCSHGC). In this choice of gauge, the complete Einstein-Yang-Mills equations
reduce to a coupled elliptic-hyperbolic system. Utilizing the method developed
by Andersson and Moncrief \cite{andersson2003elliptic}, we establish the
existence of a unique, local, continuous-in-time solution of this coupled
system. This yields an `in time' continuation criteria of the solutions which
is to be used in the potential future proof of an improved continuation
criteria for this coupled system utilizing Moncrief's light cone estimate
technique.
