In quantum mechanics monodromy occurs when there are no good quantum numbers. For instance, when a linear molecule is rotating in a free space, the rotational angular momentum J is conserved, and so is its projection to the Z-axis, M. When we turn on an electric field, the space isn't isotropic anymore, and the angular momentum J is not conserved, while its projection to the field direction, M, is. Then, if another field is applied, noncollinear with the first one, no good quantum numbers will remain to describe the molecule. In such a case there are no "good" ways to label molecular states. Also, a strange phenomenon, called "the label switching" appears: labels of states depend on in which order you turn on two noncollinear fields.
When everything goes bad monodromy appears on the scene: this is the case for a molecule in two tilted fields (article with preprint).
Coming back to preprint by Fitch et al, which will appear in PRL soon: this is the first observation of monodromy in a classical system ever. And it's a pleasure to read such a good-written article by guys, usually working in the extreme quantum regime.