To make this into a separate question:

*If the supports of a sequence of complex algebraic curves in $\mathbb{CP}^n$ (images of non-constant holomorphic maps from compact Riemann surfaces) converge to a real compact immersed submanifold $M \subset \mathbb{CP}^n$, must this submanifold $M$ be complex (algebraic)?*

For some motivation for this problem, see here:

https://mathoverflow.net/questions/176629/if-there-is-a-dense-geodesic-are-almost-all-geodesics-equidistributed-dense

Note that if $M$ is complex algebraic, there is certainly a sequence of complex algebraic curves with supports converging to $M$. (In fact, they could be taken lie in $M$). What I ask about is the converse. Unless it has an easy counterexample, this question is somewhat in the spirit of a few famous problems about special subvarieties in diophantine geometry, such as the Manin-Mumford and Andre-Oort questions, or Lang's algebro-geometric conjecture.