Extension theorem over reals

Question

Is there an equivalent of the following theorem from Cox, Little & O'Shea over reals?

Definition. Given $I=\left\langle f_{1}, \ldots, f_{s}\right\rangle \subseteq k\left[x_{1}, \ldots, x_{n}\right]$, the $l$-th elimination ideal $I_{l}$ is the ideal of $k\left[x_{l+1}, \ldots, x_{n}\right]$ defined by $$ I_{l}=I \cap k\left[x_{l+1}, \ldots, x_{n}\right] $$

The Extension Theorem. Let $I=\left\langle f_{1}, \ldots, f_{s}\right\rangle \subseteq \mathbb{C}\left[x_{1}, \ldots, x_{n}\right]$ and let $I_{1}$ be the first elimination ideal of $I$. For each $1 \leq i \leq s$, write $f_{i}$ in the form $$ f_{i}=c_{i}\left(x_{2}, \ldots, x_{n}\right) x_{1}^{N_{i}}+\text { terms in which } x_{1} \text { has degree }<N_{i} $$ where $N_{i} \geq 0$ and $c_{i} \in \mathbb{C}\left[x_{2}, \ldots, x_{n}\right]$ is nonzero. Suppose that we have a partial solution $\left(a_{2}, \ldots, a_{n}\right) \in \mathbf{V}\left(I_{1}\right)$. If $\left(a_{2}, \ldots, a_{n}\right) \notin \mathbf{V}\left(c_{1}, \ldots, c_{s}\right)$, then there exists $a_{1} \in \mathbb{C}$ such that $\left(a_{1}, a_{2}, \ldots, a_{n}\right) \in \mathbf{V}(I)$.

Question. Let $I=\left\langle f_{1}, \ldots, f_{s}\right\rangle \subseteq \mathbb{R}\left[x_{1}, \ldots, x_{n}\right]$ and and let $I_{1}$ be the first elimination ideal of $I$. Suppose I have a real partial solution $\left(a_{2}, \ldots, a_{n}\right) \in \mathbf{V}\left(I_{1}\right)$, i.e., $\left(a_{2}, \ldots, a_{n}\right) \in \mathbb{R}^{n-1}$.

1. How can I tell whether there exists $a_1 \in \mathbb{R}$ such that $\left(a_{1}, a_{2}, \ldots, a_{n}\right) \in \mathbf{V}(I)$?
2. At the moment, I am finding $\left(a_{2}, \ldots, a_{n}\right) \in \mathbf{V}\left(I_{1}\right)$ numerically. Is there a formal algebraic way to do it?

I can use Hilbert's Nullstellensatz to check whether $-1 \in I$ and $-1 \in I_1$ to see whether there are any solutions over complex numbers. But what if $-1 \notin I$ and $-1 \notin I_1$ but there are no real points on $\mathbf{V}\left(I_{1}\right)$ and $\mathbf{V}\left(I\right)$. Or if the real point $\left(a_{2}, \ldots, a_{n}\right) \in \mathbf{V}\left(I_{1}\right)$ does not extend to a real point in $\mathbf{V}\left(I\right)$?

Book. Cox, David, John Little, and Donal OShea. Ideals, varieties, and algorithms: an introduction to computational algebraic geometry and commutative algebra. Springer Science & Business Media, 2013.