We present calculations of the transmutation of initially pure tungsten first-wall and divertor plasma-facing armor into W-Re-Os alloys in the European Union Power Plant Conceptual Study (PPCS) fusion plant models A, B, and AB. The fusion neutron spectrum was modeled using the MCNP Monte Carlo code including resonance self-shielding effects, and we have calculated the evolution of the W-Re-Os alloy compositions. Trajectories of the alloys in the thermodynamic phase diagram show that the alloys remain in the single body-centered-cubic phase for their service lifetimes. Results for PPCS models A and B with soft neutron spectra show that the first-wall armor transmutes to an end-of-service alloy composition of approximately 91 at.% tungsten, 6 at.% rhenium, and 3 at.% osmium at its rear face. On the plasma-facing side of the tungsten, the effect of neutron shielding is larger. For PPCS model AB, the neutron spectrum is energetically harder, resulting in significantly lower tungsten transmutation rates.