Recent technological advances in many domains including both genomics and brain imaging have led to an abundance of high-dimensional and correlated data being routinely collected. Classical multivariate approaches like Multivariate Analysis of Variance (MANOVA) and Canonical Correlation Analysis (CCA) can be used to study relationships between such multivariate datasets. Yet, special care is required with high-dimensional data, as the test statistics may be ill-defined and classical inference procedures break down. In this work, we explain how valid p-values can be derived for these multivariate methods even in high dimensional datasets. Our main contribution is an empirical estimator for the largest root distribution of a singular double Wishart problem; this general framework underlies many common multivariate analysis approaches. From a small number of permutations of the data, we estimate the location and scale parameters of a parametric Tracy-Widom family that provides a good approximation of this distribution. Through simulations, we show that this estimated distribution also leads to valid p-values that can be used for high-dimensional inference. We then apply our approach to a pathway-based analysis of the association between DNA methylation and disease type in patients with systemic auto-immune rheumatic diseases.