  We prove that if X, X' are closed subschemes of a torus T over a
non-Archimedean field K, of complementary codimension and with finite
intersection, then the stable tropical intersection along a (possibly
positive-dimensional, possibly unbounded) connected component C of Trop(X) \cap
Trop(X') lifts to algebraic intersection points, with multiplicities. This
theorem requires potentially passing to a suitable toric variety X(\Delta) and
its associated extended tropicalization N_R(\Delta); the algebraic intersection
points lifting the stable tropical intersection will have tropicalization
somewhere in the closure of C in N_R(\Delta). The proof involves a result on
continuity of intersection numbers in the context of non-Archimedean analytic
spaces.
