In subatomic physics, strongly interacting multi-hadron systems are becoming increasingly relevant. In flavor physics, for example, one can now study heavy meson decays into three hadrons to extract the Cabibbo-Kobayashi-Maskawa matrix elements, and to search for New Physics beyond the Standard Model. The success of those analyses strongly depends on the availability of rigorous models for reactions involving three-hadron final states, which are currently lacking. This decade it has become possible to compute hadronic interactions directly from Quantum Chromodynamics (QCD), using Lattice QCD. Numerical lattice calculations on three-body interactions are expected in the near future, but they hinge on the availability of three-body reaction amplitudes. In the past, strong three-body interactions have been modeled via a factorized two-body interaction scheme, where one of the particles is assumed to be a spectator. This approach is insufficient for the high-precision applications that are possible today. In this project, we propose a scheme to develop amplitudes for the full three-body interaction. This will aid the first extraction of three-particle resonances from ab-initio QCD calculations, and will allow for a far more reliable determination of the CP violating phases. In this way, one can bring down the hadronic uncertainties of New Physics searches involving multi-particle final states.