We develop a far-from-equilibrium hydrodynamic model to evolve ultrarelativistic heavy-ion collisions in event-by-event simulations. Anisotropic hydrodynamics is designed to better handle the strong and highly anisotropic expansion during the early stages of the collision. The large gradients cause conventional second-order viscous hydrodynamic approaches to break down at early times. Anisotropic hydrodynamics evolves the large pressure anisotropies present in the quark-gluon plasma non-perturbatively, which prevents negative longitudinal pressures from developing even under extreme conditions. This increased stability allows us to start anisotropic hydrodynamics already at a very early longitudinal proper time to evolve the pre-hydrodynamic stage. In current pre-hydrodynamic models, the equation of state is not consistent with the QCD equation of state used in the subsequent fluid dynamic stage. Since our approach avoids this inconsistency, we are able to achieve a smooth transition to non-conformal viscous hydrodynamics as the gradients decrease over time. For our first phenomenological application, we apply our new simulation to model fluctuating Pb+Pb collisions at LHC energies and find that our preliminary calculations for the hadronic observables are in excellent agreement with the experimental data.