We report the first three-dimensional, global-scale numerical simulations of stably stratified turbulence in the solar tachocline driven by penetrative convection. We adopt a thin-shell geometry and approximate the effects of penetrative convection via random, small-scale forcing. After the simulations have been evolved for some time, a differential rotation is introduced and maintained against viscous dissipation by an additional forcing term. The imposed shear is stable over the timescales considered (Richardson numbers range from ~250 to 1000). We investigate the characteristics of the turbulence before and after adding the imposed differential rotation. The random forcing is varied to generate different types of turbulent wave fields, dominated by rotational r-modes or irrotational g-modes. Nonlinear transfer among modes does occur, but we find little evidence for a quasi-two-dimensional inverse cascade. Coupling between the randomly forced turbulence and the imposed shear flow gives rise to Reynolds stresses that transport momentum poleward and outward. This implies diffusive (down-gradient) latitudinal transport and antidiffusive (countergradient) vertical transport. Overall, the latitudinal transport dominates and acts to dissipate the imposed shear. This could have important implications for the structure of the solar tachocline and the global angular momentum distribution in the solar interior.