Knowledge of all correlation functions of a system is equivalent to solving the corresponding many-body problem. Already a finite set of correlation functions can be sufficient to describe a quantum many-body system if correlations factorise, at least approximately. While being a powerful theoretical concept, an implementation based on experimental data has so far remained elusive. Here, this is achieved by applying it to a non-trivial quantum many-body problem: A pair of tunnel-coupled one-dimensional atomic superfluids. From measured interference patterns we extract phase correlation functions up to tenth order and analyse if, and under which conditions, they factorise. This characterises the essential features of the system, the relevant quasiparticles, their interactions and possible topologically distinct vacua. We verify that in thermal equilibrium the physics can be described by the quantum sine-Gordon model, relevant for a wide variety of disciplines from particle to condensed-matter physics. Our experiment establishes a general method to analyse quantum many-body systems in experiments. It represents a crucial ingredient towards the implementation and verification of quantum simulators.