  We propose an efficient solver for saddle point problems arising from finite
element approximations of nonlocal multi-phase Allen--Cahn variational
inequalities. The solver is seen to behave mesh independently and to have only
a very mild dependence on the number of phase field variables. In addition we
prove convergence, in three GMRES iterations, of the approximation of the two
phase problem, regardless of mesh size or interfacial width. Numerical results
are presented that illustrate the competitiveness of this approach.
