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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 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.

Addendum. I guess I'll take a stab at answering the actual question concerning sufficient conditions for real-representability. The quickest thing that I can think of is that all uniform matroids are real representable. To see this, let $U_{k,n}$ be a uniform matroid. By taking $n$ 'random' vectors in $\mathbb{R}^k$ we get a representation of $U_{k.n}$ over $\mathbb{R}$. This is a pretty rich class, and perhaps is sufficient for your purposes.

I'll also mention that the problem of testing real-representability is decidable. This follows from the Real Nullstellensatz.

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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.