Simple examples of non-abelian fermionic T-duality in supergravity

Field transformation rules of the standard fermionic T-duality require fermionic isometries to anticommute, which leads to complexification of the Killing spinors and results in complex valued dual backgrounds. We generalize the field transformations to the setting with non-anticommuting fermionic isome- tries and show that the resulting backgrounds are solutions of double field theory. Explicit examples of non-abelian fermionic T-dualities that produce complex and real backgrounds in Minkowski space-time are given. Some of our examples can be bosonic T-dualized into usual supergravity solutions, while the others are genuinely non-geometric. There is sort of classification of geometric and non-geometric solutions in this case and their correspondence with dual time.