We propose a new method for unsupervised generative continual learning through realignment of Variational Autoencoder's latent space. Deep generative models suffer from catastrophic forgetting in the same way as other neural structures. Recent generative continual learning works approach this problem and try to learn from new data without forgetting previous knowledge. However, those methods usually focus on artificial scenarios where examples share almost no similarity between subsequent portions of data - an assumption not realistic in the real-life applications of continual learning. In this work, we identify this limitation and posit the goal of generative continual learning as a knowledge accumulation task. We solve it by continuously aligning latent representations of new data that we call bands in additional latent space where examples are encoded independently of their source task. In addition, we introduce a method for controlled forgetting of past data that simplifies this process. On top of the standard continual learning benchmarks, we propose a novel challenging knowledge consolidation scenario and show that the proposed approach outperforms state-of-the-art by up to twofold across all experiments and additional real-life evaluation. To our knowledge, Multiband VAE is the first method to show forward and backward knowledge transfer in generative continual learning.