We consider maximal non-$l$-intertwining collections, which are a higher-dimensional version of the maximal non-crossing collections which give clusters of Pl\"ucker coordinates in the Grassmannian coordinate ring, as described by Scott. We extend a method of Scott for producing such collections, which are related to tensor products of higher Auslander algebras of type $A$. We show that a higher preprojective algebra of the tensor product of two $d$-representation-finite algebras has a $d$-precluster-tilting subcategory. Finally we relate mutations of these collections to a form of tilting for these algebras.