We characterise the embedding of the spatial product of two Arveson systems into their tensor product using the random set technique. An important implication is that the spatial tensor product does not depend on the choice of the reference units, i.e. it is an intrinsic construction. There is a continuous range of examples coming from the zero sets of Bessel processes where the two products do not coincide. The lattice of all subsystems of the tensor product is analysed in different cases. As a by-product, the Arveson systems coming from Bessel zeros prove to be primitive in the sense of Ref. 12.