We study a generalization of Wyner’s Common Information to Watanabe’s Total Correlation. The first minimizes the description size required for a variable that can make two other random variables conditionally independent. If independence is unattainable, Watanabe’s total (conditional) correlation is measure to check just how independent they have become. Following up on earlier work for scalar Gaussians, we discuss the minimization of total correlation for Gaussian vector sources. Using Gaussian auxiliaries, we show one should transform two vectors of length d into d independent pairs, after which a reverse water filling procedure distributes the minimization over all these pairs. Lastly, we show how this minimization of total conditional correlation fits a lossy coding problem by using the Gray–Wyner network as a model for a caching problem.