In [Hartshorne, Proposition 7.3.] as well as in [Görtz & Wedhorn, Rem. 13.55] and [Vakil Notes, around 19.2] the following is said: If $X$ is a curve over (let's say) $\mathbb{C}$ (algebraically closed) and we have a complete linear system (or an invertible sheaf and generating sections) which separates points and tangent vectors, then we get a closed embedding into $\mathbb{P}^{n}_{\mathbb{C}}$.
But I would like to start with a curve over $\mathbb{R}$ and get an embedding into $\mathbb{P}^{n}_{\mathbb{R}}$. Which conditions are needed for getting a similiar statement (involving complete linear systems separating points and tangent vectors) in the real case?
For example: My curves are nonsingular, integral, projective, schemes over $\mathbb{R}$ of dimension one.
Is there a further references besides the ones from above? Thank you!