We propose the first computationally tractable framework to solve multi-stage stochastic optimal power flow (OPF) problems in alternating current (AC) power systems. To this end, we use recent results on dual convex semi-definite programming (SDP) relaxations of OPF problems in order to adapt the stochastic dual dynamic programming (SDDP) algorithm for problems with a Markovian structure, employing scenario lattices to discretize the underlying randomness. We show that the usual SDDP lower bound remains valid and that the algorithm converges to a globally optimal solution of the stochastic AC-OPF problem as long as the SDP relaxations are tight. To test the practical viability of our approach, we set up an extensive case study of a storage sitting, sizing, and operations problem under uncertainty about demand and renewable generation using the IEEE RTS-GMLC network. We show that the convex SDP relaxation of the stochastic problem is usually tight and discuss ways to obtain near-optimal physically feasible solutions when this is not the case. Using these results, we demonstrate that the algorithm finds a physically feasible policy with a small optimality gap to the original non-convex problem and yields a significant added value of 27\% over a rolling deterministic planning policy.