We extend the Mixed Quantum-Classical Initial Value Representation (MQC-IVR), a semiclassical method for computing real-time correlation functions, to electronically nonadiabatic systems using the Meyer-Miller-Stock-Thoss (MMST) Hamiltonian in order to treat electronic and nuclear degrees of freedom (dofs) within a consistent dynamic framework. We introduce an efficient symplectic integration scheme, the MInt algorithm, for numerical time evolution of the phase space variables and monodromy matrix under the non-separable MMST Hamiltonian. We then calculate the probability of transmission through a curve crossing in model two-level systems and show that MQC-IVR reproduces quantum-limit semiclassical results in good agreement with exact quantum methods in one limit, and in the other limit yields results that are in keeping with classical limit semiclassical methods like linearized IVR. Finally, exploiting the ability of the MQC-IVR to quantize different dofs to different extents, we present a detailed study of the extents to which quantizing the nuclear and electronic dofs improves numerical convergence properties without significant loss of accuracy.