Let $K$ be a valued field and let $L \subset K^n$ be an $r$-dimensional linear subspace. For each $r$-subset $B$, let $p(B)$ be the valuation of the Plucker coordinate indexed by $B$, with $p(B)=\infty$ if that coordinate vanishes. Let $\mathcal L(p)$ denote the set of all $w\in\mathbb R^n$ such that, for every valuated circuit $c$ of $p$, the minimum of $c_i+w_i$ over $i\in[n]$ is attained at least twice. With $\operatorname{Trop}(L)$ taken inside the torus $(K^\times)^n$ and then closed in $\mathbb R^n$, allowing valued-field extensions when taking lifts, $p$ is a valuated matroid and $\operatorname{Trop}(L)=\mathcal L(p)$.