Obstruction for a real algebraic surface to be a complex algebraic curve Not every real algebraic surface can be endowned a structure of a complex algebraic curve. The only obstruction I know is orientability. 
Are there any others?
 A: As observed in some of the previous comments, every closed (=compact, without boundary) orientable real 2-manifold admits a complex structure. So in the smooth case orientability is essentially the only obstruction.
If one also considers the case of singular real algebraic surfaces, the situation is more involved and I don't know whether satisfactory results are known.
Anyway, one obvious obstruction is the presence of non-isolated singularities, since every complex curve has only a finite number of singular points.
For instance, take $X:=S^1 \times C$, where $C \subset \mathbb{RP}^2$ is the nodal real cubic of equation $y^2z=x^3+x^2z$. The singular locus of $X$ is isomorphic to $S^1$, so $X$ surely cannot be endowed with the structure of a complex algebraic curve.
A: If you're not requiring any compatibility criterion between the real and complex structure, then the only obstruction is in fact orientability.  Every smooth projective real algebraic surface is a smooth compact real 2-manifold (without boundary).  If it's orientable, it must then be a surface of genus $g$ for some $g$.  But every surface of genus $g$ admits a complex structure, and every Riemann surface is algebraic.
I don't study real algebraic geometry much, but I'm not aware of a good compatibility condition to impose on your complex structure.  If you've got something in mind, let me know.
A: If it is orientable, you have a complex structure and the field of meromorphic functions.
Putting my ears into the firing line, I suggest that something should go wrong with the transcendence degree of the field of meromorphic functions. If it is 1, you can consider DVR-s that will give you a compact algebraic curve, and I see no reason for the original curve not to be a subset. If it is more than 1 the surface cannot be algebraic...
