Is it possible to transform my variables to their ranks and apply MANCOVA to the transformed data, just like how it is done in rank ANCOVA?
In this case Peter O'Brien's method of turning the problem around makes sense. Use a polytomous logistic regression model to estimate the joint relationship between the two dependent variables and the probability of being assigned to each of the three training methods. This makes no distributional assumptions. You can relax assumptions that are akin to the equal covariance matrix assumption by including nonlinear terms on the right hand side of the model. See http://www.citeulike.org/user/harrelfe/article/13264639