We propose an integrability setup for the computation of correlation functions of gauge-invariant operators in N=4 supersymmetric Yang-Mills theory at higher orders in the large N_{c} genus expansion and at any order in the 't Hooft coupling g_{YM}^{2}N_{c}. In this multistep proposal, one polygonizes the string world sheet in all possible ways, hexagonalizes all resulting polygons, and sprinkles mirror particles over all hexagon junctions to obtain the full correlator. We test our integrability-based conjecture against a nonplanar four-point correlator of large 1/2 Bogomol'nyi-Prasad-Sommerfield operators at one and two loops.