  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
isometries and show that the resulting backgrounds are solutions of double
field theory. Explicit examples of non-abelian fermionic T-dualities that
produce real backgrounds are given. Some of our examples can be bosonic
T-dualized into usual supergravity solutions, while the others are genuinely
non-geometric. Comparison with alternative treatment based on sigma models on
supercosets shows consistency.
