Translating real portrait video into anime is an application of interest to both consumers and researchers. However, anime differs considerably from portraits, making portrait-to-anime translation challenging. Existing StyleGAN-based portrait stylization works assume that the portrait and stylized generators share the same latent space, but this assumption fails in the style of anime due to the large domain gap. Moreover, directly applying them to each video frame often leads to undesirable temporal inconsistencies. In this paper, we argue that two latent spaces with a large domain gap cannot be shared but can be related by a transformation, and develop a cyclic transformation network to connect the two spaces with two cycle constraints. This provides high-quality translation for each frame. We extend our framework to video transformation by proposing a novel frame interpolation constraint which ensures that in-between frames can be interpolated from their neighboring frames, guaranteeing temporal coherence across translated frames. Together with latent code smoothing regularization, this provides temporally coherent video-to-anime translation. Extensive experiments demonstrate that our framework outperforms state-of-the-art methods both qualitatively and quantitatively.