Quantum simulators have recently enabled experimental observations of quantum many-body systems’ internal thermalisation. Often, the global energy and particle number are conserved, and the system is prepared with a well-defined particle number—in a microcanonical subspace. However, quantum evolution can also conserve quantities, or charges, that fail to commute with each other. Noncommuting charges have recently emerged as a subfield at the intersection of quantum thermodynamics and quantum information. We initiate the experimental testing of its predictions, with a trapped-ion simulator. We prepare 6–15 spins in an approximate microcanonical subspace, a generalisation of the microcanonical subspace for accommodating noncommuting charges, which cannot necessarily have well-defined nontrivial values simultaneously. The noncommuting charges are the three spin components. We simulate a Heisenberg evolution using laser-induced entangling interactions and collective spin rotations. We report the first experimental observation of a novel non-Abelian thermal state, predicted by quantum thermodynamics. We observe reduced many-body thermalization in the presence of noncommuting charges. Quantum non-commutation effects are detectable and significant for relatively large realistic systems, despite decoherence. This work initiates the experimental testing of a subfield that has so far remained theoretical.