A while ago I tried putting infinite polytopes (space filling tessalations) into cartensian product but failed! recently I worked it out. note I'm only showing polygons (and polygon tilings)

if you take the prism of a tiling you wouyld expect it to be layers of the tessalation but in fact the prism only has one layer.

The infinite layer is the CP of the tiling with the apeirogon (line segment tessalation)

if you have duoprisms tesselations, such as 3-6 duoprisms that would be a CP of the triangle tiling and the hexagon tiling.

An infinite times an infinite is a duoprism tessalation for each piece times each piece.

An infinite times finite would be infinite in some directions but infinite in others

e.g. Triangle times hexagonal tiling is a tiling of 3-6 duoprisms, but in one layer (of the triangular girdle of hexagonal prisms) but infinite for the other girdle

This is just an idea for how to do it