If $k=\mathbb{C}$, $X$ is a projective (or even proper) scheme over $k$ and $F$ a coherent sheaf on $X$, then there is an isomorphism

$$ H^p(X,F)\rightarrow H^p(X^{an}, F^{an}) $$

(and there is also a relative version of this, for proper morphisms between $k$-schemes locally of finite type).

If take $k$ to be $\mathbb{R}$, there still is an analytification functor, so I wonder wether this result still holds and if you could provide a reference for this.

Remarks: (i) I must admit that I haven't studied the details of the proof in the complex setting (yet), so from my current knowledge it would be possible that it carries over without major modifications. If this is not the case, what goes wrong?

(ii) In case there are additional assumptions to be made, I would also be happy with a statement for smooth varieties.