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The formation of the intermetallic $\sigma$ phase (space group $P4_2/mnm$) has a detrimental effect on the ductility of transition-metal alloys. This theoretical study uses first-principles calculations to investigate the stability and thermodynamic properties of the WRe and WOs $\sigma$ phases. Our study indicates that the $\sigma$ phase becomes thermodynamically stable at its ideal composition for temperatures above 1050 K and 130 K for the WRe and WOs phases, respectively. We find that models that neglect the phonon contribution to the free energy may underestimate the amount of disorder in the $\sigma$ phase at elevated temperatures. Furthermore, the $\sigma$ phase becomes dynamically unstable for Re concentrations above 73 at.% Re and 53 at.% Os. For the WOs phase, the dynamic stability is sensitive to the lattice site occupation, and vibrations of the Os atoms are linked to the transformation into a dynamically stable orthorhombic phase.