There is a rich and growing literature on producing local contrastive/counterfactual explanations for black-box models (e.g. neural networks). In these methods, for an input, an explanation is in the form of a contrast point differing in very few features from the original input and lying in a different class. Other works try to build globally interpretable models like decision trees and rule lists based on the data using actual labels or based on the black-box models predictions. Although these interpretable global models can be useful, they may not be consistent with local explanations from a specific black-box of choice. In this work, we explore the question: Can we produce a transparent global model that is simultaneously accurate and consistent with the local (contrastive) explanations of the black-box model? We introduce a natural local consistency metric that quantifies if the local explanations and predictions of the black-box model are also consistent with the proxy global transparent model. Based on a key insight we propose a novel method where we create custom boolean features from sparse local contrastive explanations of the black-box model and then train a globally transparent model on just these, and showcase empirically that such models have higher local consistency compared with other known strategies, while still being close in performance to models that are trained with access to the original data.