The coupled Allen-Cahn/Cahn-Hilliard system consists of high-order partial differential equations that make explicit methods hard to apply due to the severe restriction on the time step size. In order to relax the restriction and obtain steady-state solution(s) in an efficient way, we use a fully implicit method for the coupled system and employ a Newton-Krylov-Schwarz algorithm to solve the nonlinear algebraic equations arising at each time step. In the Schwarz preconditioner we impose low-order homogeneous boundary conditions for subdomain problems. We investigate several choices of subdomain solvers as well as different overlaps. Numerical experiments on a supercomputer with thousands of processor cores are provided to show the scalability of the fully implicit solver.