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This paper concerns the problem of constructing joint prediction regions having coverage probability equal or close to the target nominal value. In particular, the focus is on prediction regions defined using a system of simultaneous prediction limits. These regions are not necessarily of rectangular form and each component prediction interval depends on the preceding future observations. The specification of prediction regions with well-calibrated coverage probability has been considered in Corcuera and Giummolè (2006) and Vidoni (2015). In this paper we consider an asymptotically equivalent procedure, which extends to the multivariate setting the bootstrap-based approach proposed in Uekiand Fueda (2007). A simple application to autoregressive time series models is presented.