Let $\tau:GL(n,\mathbb{R}) \rightarrow GL(V)$ be a rational representation of the general linear group of degree $n$ on a finite-dimensional real vector space $V$. Let $C$ be an irreducible real algebraic set in $V$ such that the action of $GL(n,\mathbb{R})$ on $V$ induced by $\tau$ leaves the set $C$ invariant.

We endow $C$ with the subspace topology inherited from (Hausdorf) Euclidean topology of $V$. Supose there is a point $v\in C$ such that the orbit $GL(n,\mathbb{R})⋅v$ is an open subset of $C$.

My questions is: is it true that the orbit $GL(n,\mathbb{R})⋅v$ must be a Zariski-open subset of $C$?