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](http://homepages.ecs.vuw.ac.nz/~mayhew/Publications/MNW09.pdf), 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.