At cross-purposes

cross-purpose

[kraws-pur-puhs, kros-]
noun
1.
an opposing or contrary purpose.
Idioms
2.
at cross-purposes, in a way that involves or produces mutual misunderstanding or frustration, usually unintentionally.

Origin:
1660–70

Dictionary.com Unabridged
Based on the Random House Dictionary, © Random House, Inc. 2014.
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World English Dictionary
cross-purpose
 
n
1.  a contrary aim or purpose
2.  at cross-purposes conflicting; opposed; disagreeing

Collins English Dictionary - Complete & Unabridged 10th Edition
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