Real Dictionary

at cross-purposes
phrase: If people are at cross-purposes, they do not understand each other because they are working towards or talking about different things without realizing it.

at cross-purposes in British English
conflicting; opposed; disagreeing
having, or acting under, a misunderstanding as to each other's purposes
if two people are at cross purposes, there is a misunderstanding between them because they think they are talking about or trying to do the same thing as each other, but they are actually talking about or trying to do different things
in a way that involves or produces mutual misunderstanding or frustration, usually unintentionally
not understanding; confused