The more general statement that is true is the following:

Let $X \subset \mathbb{C}^d$ be an irreducible affine variety defined by real polynomials. If $X$ has a smooth real point, then $X(\mathbb{R})$ is Zariski dense in $X.$

A sketch of the proof and some good examples are given here: Real Algebraic Geometry for Geometric Constraints by Frank Sottile (page 8).

The main reference is: Real Algebraic Geometry by Bochnak, Coste, Roy.

Fun facts:

1. If you ask instead for the density of $\overline{\mathbb{Q}}$ then that is true more generally for any $d$–dimensional affine variety $V$ defined over $\mathbb{Q}$ by Noether Normalization.
2. If you remove the assumption that there is a smooth point, then the statement is false as $V(x^2+y^2)$ has only one $\mathbb{R}$-point which is not smooth, namely $(0,0)$, and $x^2+y^2$ is irreducible over $\mathbb{R}$ and the $\mathbb{C}$-locus is dimension 1 (so the $\mathbb{R}$-locus not Zariski dense).

The more general statement that is true is the following:

Let $X \subset \mathbb{C}^d$ be an irreducible affine variety defined by real polynomials. If $X$ has a smooth real point, then $X(\mathbb{R})$ is Zariski dense in $X.$

A sketch of the proof and some good examples are given here: Real Algebraic Geometry for Geometric Constraints by Frank Sottile (page 8).

The main fact used is, under the assumption there is a real smooth point, the smooth points of the $\mathbb{R}$-locus is a real manifold of the same dimension.

The main reference is: Real Algebraic Geometry by Bochnak, Coste, Roy.

Fun facts:

1. If you ask instead for the density of $\overline{\mathbb{Q}}$-points then that is true more generally for any $d$–dimensional affine variety $V$ defined over $\mathbb{Q}$ by Noether Normalization.
2. If you remove the assumption that there is a smooth point, then the statement is false as $V(x^2+y^2)$ has only one $\mathbb{R}$-point which is not smooth, namely $(0,0)$, and $x^2+y^2$ is irreducible over $\mathbb{R}$ and the $\mathbb{C}$-locus is dimension 1 (so the $\mathbb{R}$-locus not Zariski dense).