I feel like we all run into this at some point. We've had this occur with people on the same functional team or across functional boundaries (ie. project manager and developer). What we've found in our organization is that trust is important and part of trust is healthy conflict.
It sounds like you might have some unhealthy conflict and when we've seen that occur we sat the two individuals down and had them discuss their differences and assisted in helping them both work on being empathetic towards the other.
Some people are more receptive to this than others, but being heavy handed in our smaller organization just didn't work as well as coaching them through resolution.
Hope that helps!
I am curious to know as well, as I am dealing with the same issue.