Dynamical maps describe general transformations of the state of a physical system—their iteration interpreted as generating a discrete time evolution. Prime examples include classical nonlinear systems undergoing transitions to chaos. Quantum mechanical counterparts show intriguing phenomena such as dynamical localization on the single-particle level. Here we extend the concept of dynamical maps to a many-particle context, where the time evolution involves both coherent and dissipative elements: we experimentally explore the stroboscopic dynamics of a complex many-body spin model with a universal trapped ion quantum simulator. We generate long-range phase coherence of spin by an iteration of purely dissipative quantum maps and demonstrate the characteristics of competition between combined coherent and dissipative non-equilibrium evolution—the hallmark of a previously unobserved dynamical phase transition. We assess the influence of experimental errors in the quantum simulation and tackle this problem by developing an efficient error detection and reduction toolbox based on quantum feedback.