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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
目撃者の証言が大きく食い違っている。
The witnesses' testimonies greatly contradict each other.
双方の主張が食い違ったまま裁判が続いた。
The trial continued with both parties' claims in conflict.
予算と実績が食い違い、原因を調査した。
The budget and actual results didn't match, so we investigated the cause.