The contribution of this work is the introduction of a multivariate circular-linear (or poly- cylindrical) distribution obtained by combining the projected and the skew-normal. We show the flexibility of our proposal, its property of closure under marginalization and how to quantify multivariate dependence. Due to a non-identifiability issue that our proposal inherits from the projected normal, a compu- tational problem arises. We overcome it in a Bayesian framework, adding suitable latent variables and showing that posterior samples can be obtained with a post-processing of the estimation algo- rithm output. Under specific prior choices, this approach enables us to implement a Markov chain Monte Carlo algorithm relying only on Gibbs steps, where the updates of the parameters are done as if we were working with a multivariate normal likelihood. The proposed approach can be also used with the projected normal. As a proof of concept, on simulated examples we show the ability of our algorithm in recovering the parameters values and to solve the identification problem. Then the proposal is used in a real data example, where the turning-angles (circular variables) and the logarithm of the step-lengths (linear variables) of four zebras are jointly modelled.