Suppose we want to make a 2D tiling with regular (convex) polygons. The sum of the angles around a vertex must be exactly 360°. There are only a few combinations of polygons that satisfy this condition:

60° + 128.5714° + 171.4286° (3.7.42)

60° + 135° + 165° (3.8.24)

60° + 140° + 160° (3.9.18)

60° + 144° + 156° (3.10.15)

60° + 150° + 150° (3.12.12)

90° + 108° + 162° (4.5.20)

90° + 120° + 150° (4.6.12)

90° + 135° + 135° (4.8.8)

108° + 108° + 144° (5.5.10)

120° + 120° + 120° (6.6.6)

60° + 60° + 90° + 150° (3.3.4.12, 3.4.3.12)

60° + 60° + 120° + 120° (3.3.6.6, 3.6.3.6)

60° + 90° + 90° + 120° (3.4.4.6, 3.4.6.4)

90° + 90° + 90° + 90° (4.4.4.4)

60° + 60° + 60° + 60° + 120° (3.3.3.3.6)

60° + 60° + 60° + 90° + 90° (3.3.3.4.4, 3.3.4.3.4)

60° + 60° + 60° + 60° + 60° + 60° (3.3.3.3.3.3)

Many of these configurations cannot be extended to a complete tiling: Going around a triangle or a pentagon, the polygons must alternate between the other two types (for example 4-gons and 20-gons); but this is impossible because 3 and 5 are odd. Thus, only 3, 4, 6, 8, and 12-gons can be used in a tiling. If there's an octagon anywhere, it must be the uniform 4.8.8 tiling. Anything else is essentially a tiling of triangles and squares; hexagons can be "augmented with hexagonal pyramids" (or cut into triangles), and dodecagons can be "augmented with hexagonal cupolas" (or cut into triangles, squares, and hexagons).

Now let's consider 3D honeycombs with CRF cells. This has been mentioned here and here; especially significant is the cube-doe-bilbiro honeycomb.

The first task is to find combinations of dihedral angles that sum to 360°.

I started with large prisms and antiprisms (in pairs sharing a large face), systematically adding up angles with other CRF polyhedra. One difficult case to consider was two (n-gon) antiprisms meeting a third (m-gon) antiprism at two of its triangle faces: The first two dihedral angles, between a triangle and the n-gon, have a sum slightly greater than 180°, and the third angle, between the two triangles, is slightly less than 180°, so the sum is arbitrarily close to 360° (both below it and above it) when n and m are large enough. It should be possible to rule out this case using algebraic number theory, but I just noted that the m-gon antiprism must be paired with another m-gon antiprism or prism at the same vertex, and the four large solid angles (each slightly less than 180°) don't leave enough space for anything else at the vertex.

Then I wrote a program to check the finitely many remaining angles. Here are my findings:

Nothing fits with an n-gon antiprism with n=4 or n≥6. Prisms are more compatible, but the large ones, including 7,9,10,11,≥13, can't be used in a honeycomb for the same reason as in 2D. A 12-gon prism can only appear in a 2D tiling stacked on top of itself.

As you probably expected, the crown jewels J84-90 and snic and snid don't fit with anything. Also the augmented pentagonal prism (angle 162.74°) doesn't fit.

From the polyhedra with 3,4,5,6,8,10-gons, there are 55 combinations of 3 angles adding to 360°, 128 combinations of 4 angles, 112 of 5, 56 of 6, 16 of 7, 3 of 8, and no combinations of 9 or more angles.

Some dihedral angles sum to 360° but there's no way to make the faces match. This rules out the augmented tridiminished icosahedron (angle 171.34°), the biaugmented triangular prism (angle 169.47°), and the gyrate rhombicosidodecahedron and relatives (angle 153.43°).

The edge between the hexagon and a square in thawro can be completed (for example, with another thawro in gyro orientation, and a pocuro), but the vertex cannot be completed (the vertical edge of the square has a remaining angle 41.81°). So, while J91 appears in a honeycomb, J92 does not.

Similarly a vertex of grid (10.6.4) cannot be completed.

Is there any CRF honeycomb involving 10-gons?

I did find a complete vertex configuration with 10-gons; it has an augmented truncated dodecahedron, a pentagonal pyramid, a diminished rhombicosidodecahedron, and two pentagonal rotundas. I don't know whether this can be extended.

As of now, this search for vertex configurations is being done manually. It would be harder to automate than the dihedral angle sums....