We present a method for building high-fidelity animatable 3D face models that can be posed and rendered with novel lighting environments in real-time. Our main insight is that relightable models trained to produce an image lit from a single light direction can generalize to natural illumination conditions but are computationally expensive to render. On the other hand, efficient, high-fidelity face models trained with point-light data do not generalize to novel lighting conditions. We leverage the strengths of each of these two approaches. We first train an expensive but generalizable model on point-light illuminations, and use it to generate a training set of high-quality synthetic face images under natural illumination conditions. We then train an efficient model on this augmented dataset, reducing the generalization ability requirements. As the efficacy of this approach hinges on the quality of the synthetic data we can generate, we present a study of lighting pattern combinations for dynamic captures and evaluate their suitability for learning generalizable relightable models. Towards achieving the best possible quality, we present a novel approach for generating dynamic relightable faces that exceeds state-of-the-art performance. Our method is capable of capturing subtle lighting effects and can even generate compelling near-field relighting despite being trained exclusively with far-field lighting data. Finally, we motivate the utility of our model by animating it with images captured from VR-headset mounted cameras, demonstrating the first system for face-driven interactions in VR that uses a photorealistic relightable face model.