We describe a new method for simulating nonadiabatic dynamics using stochastic trajectories. The method, which we call quantum trajectory surface hopping (QTSH), is a variant of the popular fewest-switches surface-hopping (FSSH) approach, but with important differences. We briefly review and significantly extend our recently described consensus surface-hopping (CSH) formalism, which captures quantum effects such as coherence and decoherence via a collective representation of the quantum dynamics at the ensemble level. Using well-controlled further approximations, we derive an independent trajectory limit of CSH that recovers the FSSH stochastic algorithm but rejects the ad hoc momentum rescaling of FSSH in favor of quantum forces that couple classical and quantum degrees of freedom and lead to nonclassical trajectory dynamics. The approach is well-defined in both the diabatic and adiabatic representations. In the adiabatic representation, the classical dynamics are modified by a quantum-state-dependent vector potential, introducing geometric phase effects into the dynamics of multidimensional systems. Unlike FSSH, our method obeys energy conservation without any artificial momentum rescaling, eliminating undesirable features of the former such as forbidden hops and breakdown of the internal consistency of quantum and ensemble-based state probabilities. Corrections emerge naturally in the formalism that allow approximate incorporation of decoherence without the computational expense of the full CSH approach. The method is tested on several model systems. QTSH provides a surface-hopping methodology that has a rigorous foundation and broader applicability than FSSH while retaining the low computational cost of an independent trajectory framework.