Twisted differential K-theory is the twisted differential cohomology version K-theory: the combination of differential K-theory and twisted K-theory.
Roughly this is an extension of twisted de Rham cohomology by twisted K-theory via the twisted Chern character map.
twisted differential K-theory
Preliminary discussion included
Alan Carey, Differential twisted K-theory and applications , ESI preprint (pdf)
Alexander Kahle, Alessandro Valentino, T-Duality and Differential K-Theory
and specifically the need for twisted differential K-theory for the description of type II string theory backgrounds with their RR-fields and B-fields on orientifolds was highlighted in
A definition and proof of its pertinent properties of twisted differential cohomology in general and of twisted differential K-theory in particular is due to
(in terms of smooth spectra such as discussed at differential cohomology diagram and tangent cohesive (∞,1)-topos).