We analyze the effect of two-photon absorption on the propagation of high-order spatial optical solitons. The main focus of the paper is the splitting of the soliton bound state which has been recently observed experimentally. Using the results of the adiabatic perturbation theory and the information extracted from the Zakharov-Shabat eigenvalue problem we investigate the evolution of the soliton amplitudes and demonstrate that such a splitting can be understood as resulting from a sequence of the amplitude jumps which finally lead to a structural bifurcation. Different regimes of soliton splitting and decay are analyzed in detail.