Summary: Tension-compression splits are frequently introduced to remedy the symmetric behavior of phase-field models of fracture. These splits can suffer from an overly stiff model response. We propose a gradient-extended damage model with crack orientation director which locally represents the kinematics of fracture modes I, II and III. The kinematics of the model are asymmetric in tension and compression. Other authors which follow this approach are often times concerned with cleavage systems with well-known cleavage planes. In our model the crack surface orientation is not known a priori and thus treated as a degree of freedom. The formulation within the generalized standard material framework yields thermodynamic consistency. Two outstanding features of our approach are the straightforward implementation of anisotropic damage behavior and the traction free crack surface which emerges from the model. We demonstrate our findings with numerical examples for different load cases, different sample geometries and isotropic as well as anisotropic damage behavior. We showcase advantages of our approach when compared to a model which features a spectral tension-compression split.