 Solution

Let $q:N_{\mathbb{R}}\to N(\tau {)}_{\mathbb{R}}=N_{\mathbb{R}}/\mathrm{span}(\tau )$ be the quotient map. The images $q(\sigma )$ for $\tau ⪯\sigma$ satisfy the face and intersection axioms, so $\mathrm{Star}(\tau )$ is a fan.

To prove completeness, take $\bar{v}\in N(\tau {)}_{\mathbb{R}}$, choose a lift $v$, and choose $w$ in the relative interior of $\tau$. Since $\Sigma$ is complete, each $v+mw$ lies in some cone of $\Sigma$. There are finitely many cones, so one cone $\sigma$ contains $v+m_{j}w$ for an unbounded sequence $m_j$. Closedness gives $w\in \sigma$. Because $w$ lies in the relative interior of the cone $\tau$ and $\sigma \cap \tau$ is a face of $\tau$, this forces $\tau ⪯\sigma$. Finally,Thus every point of the quotient lies in the support of $\mathrm{Star}(\tau )$, proving that it is a complete fan.