Fetching details from the dictionary...

to not match well (e.g. at the seams); to not mesh properly (of gears)
to differ (e.g. of opinions); to clash; to be in conflict (e.g. with the facts); to run counter (to); to be inconsistent (with); to be contradictory
二人の証言が大きく食違うため調査が必要になった。
The two testimonies diverged greatly, necessitating an investigation.
彼女と彼の話の内容が大きく食違った。
Her account and his differed greatly.
両者の意見が根本的な点で食違っている。
The two sides' opinions are fundamentally at odds.