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:

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.