  A matroid is $\text{GF}(q)$-regular if it is representable over all proper
superfields of the field $\text{GF}(q)$. We show that, for highly connected
matroids having a large projective geometry over $\text{GF}(q)$ as a minor, the
property of $\text{GF}(q)$-regularity is equivalent to representability over
both $\text{GF}(q^2)$ and $\text{GF}(q^t)$ for some odd integer $t \geq 3$. We
do this by means of an exact structural description of all such matroids.
