A T-fold is a space that locally looks like a Riemannian manifold but is glued together from these not just by diffeomorphisms but also by T-duality transformations.
The idea is that A T-fold is a target space for a string sigma-model that is only locally a Riemannian manifold but globally a more general kind of geometry. In the literature sometimes the term non-geometric backgrounds is used for such “generalized geometric” backgrounds.
The idea was originally introduced in
Further developments are in
Chris Hull, Doubled geometry and T-folds JHEP0707:080,2007 (arXiv:hep-th/0605149)
Chris Hull, Global Aspects of T-Duality, Gauged Sigma Models and T-Folds (arXiv:hep-th/0604178)