Three dimensional Cairo Pattern

Question

The Cairo pentagonal tiling is an interesting tessellation of the two-dimensional plane by irregular pentagons, which is given by taking two irregular hexagonal tilings, congruent but perpendicular to each other, and overlaying them on top of each other. I've been doing some research into this two-dimensional tiling, and I'm wondering if there is any three-dimensional equivalent of it? Searching the literature for "three-dimensional Cairo pattern" and similar phrases has come up blank, but maybe an expert would be able to recognise the same idea under a different name. Is there any three-dimensional process, analogous to the process of forming the Cairo pattern by overlaying two hexagonal tessellations, which gives a tessellation of three-dimensional space by some irregular polyhedron?