Semiconducting alloys, in particular SiGe, have been employed for several decades as high-temperature thermoelectric materials. Devising strategies to reduce their thermal conductivity may provide a substantial improvement in their thermoelectric performance also at lower temperatures. We have carried out an ab initio investigation of the thermal conductivity of SiGe alloys with random and spatially correlated compositional disorder employing the quasiharmonic Green-Kubo (QHGK) theory with force constants computed from density functional theory. Leveraging QHGK and the hydrodynamic extrapolation to achieve size convergence, we obtained a detailed understanding of lattice heat conduction in SiGe and demonstrated that colored disorder suppresses thermal transport across the acoustic vibrational spectrum, leading to up to a fourfold enhancement in the intrinsic thermoelectric figure of merit.