Phase separation in crystalline solids is almost always accompanied by long ranged elastic stress. These stresses, which result from a difference in lattice parameter between the particle and matrix can give rise to qualitatively new phenomena as compared those in a stress-free system, such as changes in particle shape with increasing particle size, particle migration and alignment, and inverse coarsening wherein small particles grow at the expense of large particles. At issue is the manner in which these local phenomena influence the dynamics of ensemble-averaged quantities. We have investigated the dynamics of Ostwald ripening in elastically stressed crystalline solids through large-scale numerical simulations. Using the insight provided by the simulations, a theory for the dynamics of late-stage phase separation in elastically anisotropic homogeneous solids is developed. Both the theory and simulations show that for the systems considered elastic stress does not alter the exponent of the temporal power law for the average particle size but does effect the amplitude of the power law in a manner that is only a function of the symmetry of the particle morphology. The effect of interparticle elastic interactions on particle morphology and spatial correlations will also be discussed.