This does not technically answer your question, but I think it may of interest to you, so bear with me. If you are interested in excluded-minor characterizations for real-representability, the situation is in fact much worse than what Vámos proved. In this paper, Mayhew, Newman, and Whittle prove the following theorem:
Theorem. For any real-representable matroid $N$, there exists an excluded-minor for real-representability that contains $N$ as a minor.
I'll remark that the same result holds over any other infinite field. Another way to view this theorem is as follows. Let $\mathcal{R}$ be the set of real-representable matroids and let $E(\mathcal{R})$ be the set of excluded-minors for $ \mathcal{R}$. So, the theorem asserts that the downset of $E(\mathcal{R})$ contains all of $\mathcal{R}$! So, in some sense the set of excluded minors for $\mathcal{R}$ is as complicated as $\mathcal{R}$ itself. This is in striking constrast to the situation for finite fields, where Rota conjectured that the set of excluded minors is always finite.
Rota's Conjecture. For any finite field $\mathbb{F}$, the set of excluded minors for $\mathbb{F}$-representability is finite.
This conjecture has been proven for $\mathbb{F}_2, \mathbb{F}_3$, and $\mathbb{F}_4$, but is open for all other finite fields.