The occurrence of segregation in dilute alloys under irradiation is a highly unusual phenomenon that has recently attracted attention, stimulated by the interest in the fundamental properties of alloys as well as by their applications. The fact that solute atoms segregate in alloys that, according to equilibrium thermodynamics, should exhibit full solubility, has significant practical implications, as the formation of precipitates strongly affects physical and mechanical properties of alloys. A lattice Hamiltonian, generalizing the so-called 'ABV' Ising model and including collective many-body inter-atomic interactions, has been developed to treat rhenium solute atoms and vacancies in tungsten as components of a ternary alloy. The phase stability of W-Re-vacancy alloys is assessed using a combination of density functional theory (DFT) calculations and cluster expansion (CE) simulations. The accuracy of CE parametrization is evaluated against the DFT data, and the cross-validation error is found to be less than 4.2 meV/atom. The free energy of W-Re-vacancy ternary alloys is computed as a function of temperature using quasi-canonical Monte Carlo simulations, using effective two, three and four-body interactions. In the low rhenium concentration range (<5 at.[Formula: see text]Re), solute segregation is found to occur in the form of voids decorated by Re atoms. These vacancy-rhenium clusters remain stable over a broad temperature range from 800 K to 1600 K. At lower temperatures, simulations predict the formation of Re-rich rhenium-vacancy clusters taking the form of sponge-like configurations that contain from 30 to 50 at.[Formula: see text]Re. The anomalous vacancy-mediated segregation of Re atoms in W can be rationalized by analyzing binding energy dependence as a function of Re to vacancy ratio as well as chemical Re-W and Re-vacancy interactions and short-range order parameters. DFT calculations show that rhenium-vacancy binding energies can be as high as 1.5 eV if the rhenium/vacancy ratio is in the range from 2.4 to 6.6. The predicted Re clustering agrees with experimental observations of precipitation in self-ion irradiated W-2[Formula: see text] Re alloys and neutron-irradiated alloys containing 1.4 at.[Formula: see text]Re.