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Many computational problems are intractable through classical computing and, as Moore's law is drawing to a halt, demand for finding alternative methods in tackling these problems is growing. Here, we realize a liquid light machine for the NP-hard max-3-cut problem based on a network of synchronized exciton-polariton condensates. We overcome the binary limitation of the decision variables in Ising machines using the continuous-phase degrees of freedom of a coherent network of polariton condensates. The condensate network dynamical transients provide optically-fast annealing of the XY Hamiltonian. We apply the Goemans and Williamson random hyperplane technique, discretizing the XY ground state spin configuration to serve as ternary decision variables for an approximate optimal solution to the max-3-cut problem. Applications of the presented coherent network are investigated in image-segmentation tasks and in circuit design.