Is there a problem if we use non-anonymous feedback in 360?
With 360 degree feedback, it's a case of damned if you use anonymous feedback, and damned if you use non-anonymous feedback, because frankly both options have different strengths and weaknesses.
The conventional wisdom (which isn't always very wise) is that 360 feedback should be anonymous, since some people believe that anonymous feedback will be more honest, forthright and direct because people will feel protected from retribution. That can be the case. It can also be the case that feedback givers, feeling protected from the consequences of lying or providing vindictive feedback, will, in fact try to "assassinate" a person they dislike.
Given that non-anonymous feedback can be more helpful in improving performance, because it allows more direct discussion between feedback giver and receiver, perhaps the best course of action is to provide the feedback giver with a choice to be anonymous or not, and, to treat anonymous feedback differently (provide less weight or credence to it).