We extend multi-way, multivariate ANOVA-type analysis to cases where one covariate is the view, with features of each view coming from different, high- dimensional domains. The different views are assumed to be connected by having paired samples; this is common in our main application, biological experiments integrating data from different sources. Such experiments typically also include a controlled multi-way experimental setup where disease status, medical treatment groups, gender and time of the measurement are usual covariates. We introduce a multi-way latent variable model for this new task, by extending the generative model of Bayesian canonical correlation analysis (CCA) both to take multi-way covariate information into account as population priors, and by reducing the dimensionality by an integrated factor analysis that assumes the features to come in correlated groups.