On the fundamental level, quantum fluctuations or entanglement lead to complex dynamical behaviour in many-body systems for which a description as emergent phenomena can be found within the framework of quantum field theory. A central quantity in these efforts, containing all information about the measurable physical properties, is the quantum effective action. Though non-equilibrium quantum dynamics can be exactly formulated in terms of the quantum effective action, finding solutions is in general beyond the capabilities of classical computers. Here, we present a strategy to determine the non-equilibrium quantum effective action using analogue quantum simulators, and demonstrate our method experimentally with a quasi-one-dimensional spinor Bose gas out of equilibrium. Spatially resolved snapshots of the complex-valued transversal spin field allow us to infer the quantum effective action up to fourth order in an expansion in one-particle irreducible correlation functions at equal times. We uncover a strong suppression of the irreducible four vertex emerging at low momenta in the highly occupied regime far from equilibrium where perturbative descriptions fail. Our work constitutes a new realm of large-scale analogue quantum computing, where highly controlled synthetic quantum systems10 provide the means for solving theoretical problems in high-energy and condensed-matter physics with an experimental approach.