Let $m,n \in \mathbb{N}$, and let $A \in \{-1,0,1\}^{m \times n}$ be a totally unimodular integer matrix, meaning that every square minor of $A$ has determinant in $\{-1,0,1\}$. Let $E := \{1,\dots,n\}$ index the columns of $A$, and let $M(A)$ be the column matroid whose independent sets are the subsets $I \subset E$ for which the columns of $A$ indexed by $I$ are linearly independent over $\mathbb{Q}$. Then $M(A)$ is regular: for every field $F$, the matrix obtained from $A$ by interpreting its entries in $F$ represents the same matroid $M(A)$.