How to navigate this situation?

I’ve recently been promoted to manager and am currently facing my first interdepartmental conflict. Several of my team members have expressed frustration with an employee from a different department who consistently makes mistakes, primarily due to a lack of attention to detail. This requires a lot of follow-up from my team to ensure tasks are completed correctly. I’ve experienced the same issues firsthand with this employee. To address this, I gathered examples of these mistakes and shared them with the manager of the other department. She thanked me for the feedback and assured me that she would work with the employee.

However, one of my team members tends to be somewhat impulsive and has been very direct with the employee when mistakes are made. This has resulted in the employee forwarding every interaction to their manager, who then sends these complaints to me. It’s creating a toxic dynamic. My team member remains professional, but their tone can sometimes come across as harsh. Personally, I believe their frustration is justified, but when the other manager sends me screenshots of these interactions, I struggle with how to respond. My instinct is to think, "If you spent the same time and energy coaching your employee, that might be more productive for everyone," but obviously, I can’t express that directly.

I’m unsure how to address this situation professionally while also supporting my employee. Any advice on how to navigate this would be greatly appreciated.