The situation may look like a personality conflict, but the symptom leads us astray. When two people are at cross-purposes, locked in disagreement, it is because we, as managers, created the conditions for the behavior we see.
Still looks like a personality conflict?
If you are in a place of worship, a temple, synagogue, sanctuary, are you likely to be loud and boisterous or quiet and reflective? If you are at a sporting event and your team just scored a goal, are you likely to be loud and boisterous or quiet and reflective? Your behavior in those two circumstances is quite different, but did your personality change?
Your behavior changed because the context changed.
Change the context, behavior follows.
Structure is the way we define the working relationships between people in our organization. Culture is that unwritten set of rules that governs our required behaviors in the work that we do together. Structure is culture. Culture is context. Change the context, behavior follows.
Be careful how you define the working relationships in your organization. Structure creates the conditions for things that look like personality conflicts.
Change the context, behavior follows, first taught to me by Gustavo Grodnitzky.