Real Twistings are 2-Line Bundles

We construct and study a bicategory of super 2-line bundles over graded Lie groupoids, providing a unified framework for geometric models of twistings of (real) K-theory. The core of our work is to exhibit a wide range of models from the literature as special cases, among them several variants of bundle gerbes (real/equivariant/Jandl), Freed-Moore's twisted groupoid extensions, Freed-Hopkins-Teleman's K-theory twistings, Moutuou's real twistings, Freed's invertible algebra bundles, and Distler-Freed-Moore's orientifold twistings.