Definition
When two or more people are at cross purposes, they misunderstand each other because they are unknowingly discussing different subjects.
During the project discussion, we seemed to be at cross purposes because I was referring to the first phase, while she was talking about the second phase.
They were arguing about what to have for dinner, but they were at cross purposes, with one wanting Italian and the other sushi.
The client and the sales team were at cross purposes during the meeting, as the client was discussing the existing product features, while the team was promoting the new product.