Let $M$ be a matroid on the set $\{1,\dots,m\}$. The Bergman fan $\tilde{B}(M)$ is defined in literature to be the set in $\mathbb{R}^{m}$ consisting of the vectors $v=(w_1,\dots,w_m)$ such that for every circuit $c\in C(M)$, the minimum $\min_{i\in c}w_i$ is attained at least twice.
It is then mostly left implicit what the fan structure on this set should be. Sometimes people mention equivalence classes formed by $v_1\sim v_1$ iff $M_{v_1}=M_{v_2}$, where $M_v$ is the matroid with as bases those bases $B$ of $M$ attaining the minimum $\min_{B\in B(M)} \sum_{b\in B}v_b$. For what I gather it is implied that these should define a fan structure on $\tilde{B}(M)$, but simple examples show that it doesn't:
Let $M=U_{3,3}$. Then there are no circuits, so the support of $\tilde{B}(M)$ is the whole space. Since there is only a single basis, there is only a single equivalence class. So the equivalence classes do not define a fan structure on $\tilde{B}(M)$.
What am I not getting here?