We present results of the first global magnetohydrodynamic simulations of accretion disks fed by Roche-lobe overflow, including vertical stratification, in order to investigate the roles of spiral shocks, magnetorotational instability (MRI), and the accretion stream in disk structure and evolution. Our models include a simple treatment of gas thermodynamics, with orbital Mach numbers at the inner edge of the disk M-in of 5 and 10. We find mass accretion rates to vary considerably on all timescales, with only the Mach 5 model reaching a clear quasi-stationary state. For Mach 10, the model undergoes an outside-in, magnetically driven accretion event occurring on a timescale of similar to 10 orbital periods of the binary. Both models exhibit spiral shocks inclined with respect to the binary plane, with their position and inclination changing rapidly. However, the time-averaged location of these shocks in the equatorial plane is well fit by simple linear models. MRI turbulence in the disk generates toroidal magnetic field patterns (butterfly diagrams) that are in some cases irregular, perhaps due to interaction with the spiral structure. While many of our results are in good agreement with local studies, we find some features (most notably those related to spiral shocks) can only be captured in global models such as studied here. Thus, while global studies remain computationally expensive-even as idealized models-they are essential (along with more sophisticated treatment of radiation transport and disk thermodynamics) for furthering our understanding of accretion in binary systems.