We present an algorithm for learning high-treewidth Markov networks where inference is still tractable. This is made possible by exploiting context specific independence and determinism in the domain. The class of models our algorithm can learn has the same desirable properties as thin junction trees: polynomial inference, closed form weight learning, etc., but is much broader. Our algorithm searches for a feature that divides the state space into subspaces where the remaining variables decompose into independent subsets (conditioned on the feature or its negation) and recurses on each subspace/subset of variables until no useful new features can be found. We provide probabilistic performance guarantees for our algorithm under the assumption that the maximum feature length is k (the treewidth can be much larger) and dependences are of bounded strength. We also propose a greedy version of the algorithm that, while forgoing these guarantees, is much more efficient.Experiments on a variety of domains show that our approach compares favorably with thin junction trees and other Markov network structure learners.