The following is pretty much the standard argument due, I think, to Whitney (from the proof of his embedding theorem in the "stable range"). Suppose that $V\subset {\mathbb C}^N$ is an affine complex-algebraic subset defined over the real numbers (I.e. by polynomials with real coefficients), where $dim(V)=m$ and $2m+1< N$. Consider the map
$V\times V\times {\mathbb C}\to {\mathbb C}^N$, $(x,y,t)\mapsto t(x-y)$. Due to our dimension assumptions, this map is not surjective. For the same dimension reasons, its image $W$ (a complex algebraic subvariety) does not contain ${\mathbb R}^N$ and, moreover, is "small" in any meaningful sense (its dimension is $<N$). Take any (necessarily nonzero) $v\in {\mathbb R}^N \setminus W$ and let $p: {\mathbb C}^N\to {\mathbb C}^N/Span(v)$ denote the quotient map (you can think of it as the orthogonal projection to the subspace normal to $v$ with respect to the standard inner product). Then this map is injective. It is also defined over the real numbers, hence, $p(V({\mathbb R}))$ (image of the set of real points in $V$) is contained in the set of real points of ${\mathbb C}^N/Span(v)$. I claim that the set of real points of $p(V)$ equals $p(V({\mathbb R}))$. Otherwise, there is a pair of distinct complex-conjugate points $u, v\in V$ such that $p(u)=p(v)$, contradicting injectivity of $p$. Thus, $p(V({\mathbb R}))\subset {\mathbb R}^N/Span_{\mathbb R}(v)\cong {\mathbb R}^{N-1}$ is an affine real-algebraic set.
In your case, $m=1$ and hence, we can apply the projection argument inductively until the dimension of the ambient space drops to $3$.
