An effective way to mitigate the conflict is to ask to meet with the co-worker to discuss how to communicate more effectively.
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Begin the conversation by stating something along the lines of: “There seems to be tension between the two of us, as well as differences. Even though there are differences, there is also commonality. We both want to be successful, valued and respected. Let's talk about how to create a more harmonious work relationship where we can both reach our desired goals and be successful.”
Once you are in a healthy mental and emotional state you’ll be ready to listen for understanding, instead of listening to defend or reply with a rebuttal. Focus on the future and what you can both do to create a professional, healthy and productive relationship.