Some definitions.
Let us take the signature of ring theory to consist of the function symbols $\{+,-,0,\cdot,1\}$ equipped with their usual airities, where the minus symbol represents a unary operation.
Write $\mathrm{E}$ for the set of all equations in the language of rings.
By the quasi-algebraic language of rings, let us mean the following set of formulae. $$L = E \cup \bigcup_{n \geq 1}\{\varphi_0,\ldots,\varphi_{n-1} \rightarrow \psi \mid (\forall i < n)(\varphi_i \in E) \wedge (\psi \in E)\}.$$
By the quasi-algebraic theory of real matrix rings, let us mean the set $T$ of all $q \in L$ such that for every natural number $n \geq 1$, every subring of the matrix ring $M_n(\mathbb{R})$ is a model for $q$. Actually, since the formulae of interest are downward-absolute, there is no need to consider subrings at all; in particular, $T$ can be described as the set of all $q \in L$ such that for every natural number $n \geq 1$, the matrix ring $M_n(\mathbb{R})$ is a model for $q$.
For example, the formula $xy=1 \rightarrow yx=1$ is an element of $T$.
Question. Does there exist a finite axiomatization of $T$? If so, an explicit list of axioms would be much appreciated.