This article presents a soft cylindrical multistable origami structure based on “bendy straws,” consisting of multiple conical frusta mimicking the structure of a flexible drinking straw. These frusta are connected in such a way that the whole structure is axially multistable, having a stable compressed state in which its smallest frustum is collapsed. The bendy straw structure can also be modified so that the smallest frustum collapses only partially, keeping the structure in a bent state. We studied the geometry of a similar structure consisting of polygonal frusta instead of conical ones and used this geometry to design a nonrigidly foldable origami pattern folding into a similar origami bendy straw structure. Most of the origami structures presented so far have been modeled from rigidly foldable origami patterns: These origami structures do not rely on local deformations of the sheet and cannot use it to their advantage, whereas the nonrigid origami structure presented here features multistability. We have established that this origami structure is not only axially multistable, but that it can also be kept in a bent state, thanks to the use of pop-through defects (PTDs). The origami bendy straws studied here were made from paper (with a density of 90 g/m2) bilaminated with a 42.5 μm thick plastic film. A digital dynamometer was used to study the forces required to compress and expand a single origami bendy straw, create and reverse a PTD, and bend an origami bendy straw using PTDs.