The equivalence of the category of smooth projective curves over $\mathbb C$ and the category of compact Riemann surfaces is, I believe, well documented. For example, it is mentioned on the wiki page: http://en.wikipedia.org/wiki/Algebraic_curve#Compact_Riemann_surfaces and I believe that Rick Miranda's book Algebraic Curves and Riemann Surfaces goes in to some detail on this.
So I just wondered if when we look at (compact) real manifolds, can we make any relation between them and schemes over $\mathbb R$. I am not aware of any such relation, but on the other hand I do not expect to be enough of an expert on schemes any time soon to spot it myself, and my google searches were fruitless. Apologies if it is too vague/soft, and thank you for any replies :)
EDIT - To give one specific question, are there interesting subcategories of the categories of real manifolds and schemes over $\mathbb R$ that are equivalent as categories.