Germans feel uncomfortable not only with resolutions dictated "from above", but also with those resulting in a clear winner and loser. A resolution is effective and durable only if the parties involved freely accept it. "Losers" seldom freely accept.
Americans escalate disputes. They expect an authority-driven resolution process. A true and lasting resolution is attainable only when a clear decision is made. "You win some, you lose some."
The labeling of one side "the loser" breeds shame, anger, animosity. The "loser" will seek to undermine the judgement made. The conflict is not or not fully resolved. It festers. The team suffers.
German management is easily perceived as unwilling or incapable of making the tough "judgement calls." A resolution in which all parties are winners is not a decision. The conflict is not resolved. It festers. The team suffers.
Advice to Germans
You lead Americans. Make a decision. It may not involve a compromise, but a clear winner and loser. As long as your decision and the process it was based on, is fair (just), the "loser" is not lost. Again, "You win some, you lose some. It‘s not the end of the world."
You're a German with an American boss. If you "win" the conflict, do not gloat. The next decision could go against you. When that happens, you have not lost face. Do not fear being labeled a loser for life. Conflicts of interest are commonplace. Accept the decision and move on.
Advice to Americans
You lead Germans. You have arrived at a decision. Even if there is a clear winner and loser, think carefully how you will communicate it. Prevent any kind of triumphalism on the part of the winner. Soften the blow for the "loser."
The conflict is not resolved by the decision alone. How it is communicated (perceived) influences whether it is accepted and supported. Your German manager assisted to a resolution in your favor.
Fine. Make peace with your opponent. Help him/her save face. No triumphalism. It didn't go in your favor? Make peace with your opponent. Help him/her to help you save face. But, don't accept any triumphalism.