  We investigate when a Legendrian knot in standard contact $\mathbb{R}^3$ has
a non-orientable exact Lagrangian filling. We prove analogs of several results
in the orientable setting, develop new combinatorial obstructions to
fillability, and determine when several families of knots have such fillings.
In particular, we determine completely when an alternating knot (and more
generally a plus-adequate knot) is decomposably non-orientably fillable, and
classify the fillability of most torus and 3-strand pretzel knots. We also
describe rigidity phenomena of decomposable non-orientable fillings, including
finiteness of the possible normal Euler numbers of fillings, and the
minimization of crosscap numbers of fillings, obtaining results which contrast
in interesting ways with the smooth setting.
