Let $R$ be a Henselian discrete valuation ring with field of fractions $K$. If $X$ is a smooth variety over $K$ and $G$ a torus over $K$, then we consider $X$-torsors under $G$. If $𝒳/R$ is a model of $X$ then, using a result of Brahm, we show that $X$-torsors under $G$ extend to $𝒳$-torsors under a Néron model of $G$ if $G$ is split by a tamely ramified extension of $K$. It follows that the evaluation map associated to such a torsor factors through reduction to the special fibre. In this way we can use the geometry of the special...