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The very exotic nuclear resonance systems, 10He and 13Li, are produced in proton-knockout reactions from relativistic beams of 11Li and 14Be. The experimentally determined energy and angular correlations between their decay products, He 8 + n + n and Li 11 + n + n , are analyzed using an expansion of decay amplitudes in a restricted set of hyperspherical harmonics. By considering only a small number of terms it is possible to extract the expansion coefficients directly from the experimental three-body correlations. This provides a model-independent way of getting information about the decay process, on the structure of the decaying nucleus and on the quantum characteristics of the binary subsystems. The results show that the He 8 + n + n relative-energy spectrum can be interpreted as consisting of two resonances, an I π = 0 + ground state and an excited I π = 2 + state. The Li 11 + n + n relative-energy spectrum is interpreted as an I π = 3 / 2 − ground state overlapping with excited states having a structure similar to the 2 + state in 10He but spread over several states due to the coupling to the I π = 3 / 2 − core. The 13Li data also give evidence for a contribution of a configuration where the two neutrons occupy the d-shell.