The two-way PerMANOVA in PAST uses the crossed design. The algorithm follows Anderson (2001) with a modification to allow unbalanced designs (each combination of factors can contain different numbers of values). For more information see one-way PerMANOVA, but two group columns are required (as for two-way ANOSIM).
Reference
Anderson, M.J. 2001. A new method for non-parametric multivariate analysis of variance. Austral Ecology 26:32-46.