We explore the three-dimensional properties of convective, luminous (L approximate to 10(4.5)-10(5) L (circle dot)), hydrogen-rich envelopes of red supergiants (RSGs) based on radiation hydrodynamic simulations in spherical geometry using Athena++. These computations comprise approximate to 30% of the stellar volume, include gas and radiation pressure, and self-consistently track the gravitational potential for the outer approximate to 3M (circle dot) of the simulated M approximate to 15M (circle dot) stars. This work reveals a radius, R (corr), around which the nature of the convection changes. For r > R (corr), though still optically thick, diffusion of photons dominates the energy transport. Such a regime is well studied in less luminous stars, but in RSGs, the near- (or above-)Eddington luminosity (due to opacity enhancements at ionization transitions) leads to the unusual outcome of denser regions moving outward rather than inward. This region of the star also has a large amount of turbulent pressure, yielding a density structure much more extended than 1D stellar evolution predicts. This "halo" of material will impact predictions for both shock breakout and early lightcurves of Type IIP supernovae. Inside of R (corr), we find a nearly flat entropy profile as expected in the efficient regime of mixing-length theory (MLT). Radiation pressure provides approximate to 1/3 of the support against gravity in this region. Our comparisons to MLT suggest a mixing length of alpha = 3-4, consistent with the sizes of convective plumes seen in the simulations. The temporal variability of these 3D models is mostly on the timescale of the convective plume lifetimes (approximate to 300 days), with amplitudes consistent with those observed photometrically.