Let C be a geometrically connected and geometrically integral curve over a number field K and let K' be a number field containing K. Does there exist a number field L containing K such that

-   $L \cap K' = K$, and
-   $C(L) \neq \emptyset$?

Note that the hypotheses on C are necessary -- the curve x^2 + y^2 = 0, with the origin removed, is not geometrically integral and geometrically connected, but gives a counterexample for K = Q and K' = Q(i).


Also, I can prove that this is true when C has prime gonality. It would be odd, though, for this to be a necessary hypothesis.