Multivariate functional data are becoming ubiquitous with the advance of modern technology. Multivariate functional data are substantially more complex than univariate functional data. In particular, we study a novel model for multivariate functional data where the component processes exhibit mutual time warping. That is, the component processes exhibit a similar shape but are subject to time warping across their domains. To address this previously unconsidered mode of warping, we propose new registration methodology which is based on a shift-warping model. Our method differs from existing registration methods in several major ways. Namely, instead of focusing on individual-specific registration, we focus on registering across components on a population-wide level. By doing so our proposed estimates for these shifts enjoy parametric rates of convergence and often have intuitive physical interpretations. We exemplify these interpretations by applying our methodology to the Zürich Longitudinal Growth data. We also demonstrate the conditions under which our methodology works via simulation.