While searching for CRF polychora, I stumbled upon a curious fact: the dichoral angle between the pentagonal cupola in the segmentochoron 5-prism||10-gon is exactly 144°. We can therefore glue 5 copies of this segmentochoron together at their pentagonal cupola cells, and it will close up into a polychoron (just like the 600-cell lunae, except that here it doesn't close up until it wraps around the circle twice, so it's non-convex). Sadly, the result is non-convex, but it does have all CRF cells.

The resulting polychoron has a pentagrammic arrangement around the plane of the 10-gon. All 5 copies of the segmentochoron share the same 10-gon, which becomes internal to the polychoron once it closes up; the cupolae likewise become internal to the polychoron. What's left on the surface are 5 pentagonal prisms, 25 triangular prisms, and 25 tetrahedra. It looks like some kind of pentagonal-pentagrammic duoprism augmented with some kind of pentagrammic tegum bridging augments (i.e., pentagrammic pyramid prisms bridged by pentagrammic tegums).