Let $T^2$ be a real torus as an $\mathbb{R}$-affine variety. For example
$$ T^2 = \{ (x_1,x_2,x_3,x_4) \in \mathbb{R}^4 | x_1^2 + x_2^2 -1 =0, x_3^2+x_4^2-1=0 \} .$$
Let $V$ be a real irreducible variety, let's say of dimension $\ge 3$, and let $\xi: R \to T^2$ be a $\mathbb{R}$-morphism.
Let $\bar{\xi(V)}$ be the closure of the image of $V$. I have two questions:
Is it true that, if the dimension of $\bar{\xi(V)}$ is $1$, then there exists an irreducible polynomial $f \in \mathbb{R}[X_1,X_2,X_3,X_4]$ that determines $\bar{\xi(V)}$? (Meaning the $\bar{\xi(V)}$ is the set of zeros of the polynomial f in $T^2$)
And, in case the previous is true:
Is there a classification of all such polynomials?
Since in your description you are only looking at real points, I assume that the real points of $V$ are (Zariski) irreducible and that you want $f$ to cut out the real part of the Zariski closure of the image of $V$. In this case the answer is yes.
We denote $Y$ to be the closure of the image of $V$. Let $g_1,\ldots,g_r$ generators of the vanishing ideal of $Y$. Then let $G=g_1^2+\ldots+g_r^2$. The zero set of $G$ is $Y$. Now you can choose $f$ to be any irreducible factor of $G$ which vanishes on $Y$. There is at least one such factor because $Y$ is irreducible as the image of an irreducible variety.