'toe treading' is inevitable.
the left hand not knowing what the right hand is doing is never the way forward.
no idea how this can be resolved without upsetting one or the other...
Well, the theory is that DoF is there for the long term, and the head coach is responsible for the short term. Ideally the DoF is part of the club's brickwork, a well-respected ex-player or something similar. They should also constantly talk to each other. It's fine to disagree but when they finish their discussion, they should do so with a clear decision.
That's the theory.
In practice if it isn't part of the whole culture of the club (like it is in most German clubs), it usually fails miserably.