Here’s a Sonobe creation that involves breaking the underlying geometry:
Four units (light blue, light orange, red and white) have been folded across their ribs (parallel to their pockets); the other ten units (two each of green, dark blue, pink, yellow and dark orange) are normal. However, it is possible to make some reasonable identification between this figure and a recognizable polyhedron. In particular, we can replace the four bent edges with straight line segments; the result is combinatorially equivalent to the polyhedron with eight vertices , , , , , , and . This polyhedron is four times-truncated rectangular box; it has as faces four the parallelograms AEBG, BFCG, CFDH and DEAH and the four (equilateral) triangles BEF, DEF, AGH and CGH.
(Yes, it is possible that I created this image in 15 minutes using MSPaint.) Two notable features of this polyhedron are that the number of edges is not divisble by 3 and the faces are not all equilateral triangles and squares; neither feature would be possible with the conventional Sonobe construction. A variety of other similar constructions are possible: take any conventional Sonobe construction, choose a vertex, open it up, bend the units back, and then add more units. It would be interesting to try and classify exactly what sorts of polyhedra can be constructed in this way.