Let $M$ be a loopless matroid on a finite ground set $E$, and let $\widetilde{\mathcal B}(M) \subset \mathbb{R}^E$ denote the Bergman fan defined by the circuit condition

\begin{align*} \widetilde{\mathcal B}(M)=\{w \in \mathbb{R}^E : \text{for every circuit } C \text{ of } M,\ \min_{e \in C} w_e \text{ is attained at least twice}\}. \end{align*}

For a subset $F \subset E$, let $\mathbb{1}_F \in \mathbb{R}^E$ denote its indicator vector, and let $\mathbb{1}_E$ denote the all-ones vector. Then

\begin{align*} \widetilde{\mathcal B}(M)=\bigcup_{\varnothing \subsetneq F_1 \subsetneq \cdots \subsetneq F_k \subsetneq E} \left(\mathbb{R}\mathbb{1}_E+\mathbb{R}_{\geq 0}\{\mathbb{1}_{F_1},\dots,\mathbb{1}_{F_k}\}\right), \end{align*}

where the union ranges over all finite chains of proper nonempty flats of $M$, including the empty chain. Equivalently, after adjoining $F_0=\varnothing$ and $F_{k+1}=E$, these are the flag cones associated to chains of flats ending at $E$, modulo the diagonal lineality $\mathbb{R}\mathbb{1}_E$. In particular, $\widetilde{\mathcal B}(M)$ depends only on the lattice of flats of $M$.