at cross purposes

used to describe a situation in which people or groups have conflicting goals or intentions

The idiom "at cross purposes" originated from the literal idea of two or more people or groups having conflicting or opposing aims or intentions. The phrase emphasizes a lack of alignment or coordination between individuals or parties, resulting in misunderstandings or ineffective communication. It suggests that the involved parties are unintentionally working against each other, pursuing different goals or objectives that hinder cooperation or progress.

The team members were talking at cross purposes, as each had a different understanding of the project's objectives.
The couple's argument arose from a miscommunication—they were speaking at cross purposes, not truly understanding each other's perspectives.
The two political parties were discussing the issue at cross purposes, leading to a stalemate in finding a common solution.
The department heads were working at cross purposes, implementing conflicting strategies that caused confusion among the employees.
The negotiation failed because the parties involved were talking at cross purposes, unable to reach a mutual agreement.