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?



Obstruction for a real algebraic surface to be a complex algebraic curve
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.


