There is in fact a definition of such sets, they are called the set of real points of an algebraic set. Let $I \subset \mathbb{R}[x_1,\dots,x_n]$ a polynomial ideal, let $V(I)$ the algebraic set defined by $I$ over $\mathbb{C}$ and write $V_\mathbb{R}(I) = V(I)\cap \mathbb{R}^n$. The affine algebra $A = \mathbb{R}/I$ is called the $\mathbb{R}$-coordinate ring of $V_\mathbb{R}(I)$. We have the following theorem, known as the Weak Real Nullstellensatz
Theorem. $A$ is semireal if and only if $V_\mathbb{R}(I)\neq \emptyset$
The theorem is valid over a more general context, the notion of a real closed set wich is the starting point of Real Algebraic Geometry, you can check this theory on the book of Bochnak, Coste and Roy Real Algebraic Geometry.