We develop and illustrate a method for the joint planning of natural gas and electric power systems that are subject to spatially correlated failures of the kind that would be expected to occur in the case of extreme weather events. Our approach utilizes a two-stage stochastic planning and operations framework for a jointly planned and operated gas and electric power transmission system. Computational tractability is achieved through convex relaxations of the natural gas flow equations and the use of a machine learning algorithm to reduce the set of possible contingencies. We illustrate the method using a small test system used previously in the literature to evaluate computational performance of joint gas-grid models. We find that planning for geographically correlated failures rather than just random failures reduces the level of unserved energy relative to planning for random (spatially uncorrelated failures). Planning for geographically correlated failures, however, does not eliminate the susceptability of the joint gas-grid system to spatially uncorrelated failures.