A graph $G$ has an associated multimatroid $\mathcal{Z}_3(G)$, which is equivalent to the isotropic system of $G$ studied by Bouchet. In previous work it was shown that $G$ is a circle graph if and only if for every field $\mathbb F$, the rank function of $\mathcal{Z}_3(G)$ can be extended to the rank function of an $\mathbb F$-representable matroid. In the present paper we strengthen this result using a multimatroid analogue of total unimodularity. As a consequence we obtain a characterization of matroid planarity in terms of this total-unimodularity analogue.