Suppose there is a closed analytic curve $C$ on a Riemann surface $S$, that is the image of a map $\gamma$ from the equator $E$ of the Riemann sphere to the surface $S$ which is a restriction of a one-to-one complex analytic map $\Gamma$ of annulus $A(\supset E)$ into the surface $S$. Suppose that I change the complex structure on $S$: under which conditions the curve $C$ remains analytic?
On a neighborhood of $C$, the two complex structures are related by a diffeomorphism $h$, $J'=h^*J$. If $C$ is analytic with respect to $J$, it is analytic with respect to $J'$ (in the sense you described at the beginning) if and only if the restriction of $h$ to $C$ is real-analytic with respect to the analytic structure induced by $J$. One direction is clear; the extension of the map to the annulus for the opposite direction is provided by (3.4) in math/1301.1074.