This paper discusses novel coherent beam-beam instability in collisions with a large crossing angle. The instability appears in the correlated head-tail motion of two colliding beams. A cross-wake force, which is localized at the collision point, is introduced to represent the head-tail correlation between colliding beams. A mode-coupling theory based on this localized cross-wake force enables us to explain the correlated heal-tail instability. The use of a collision scheme with a large crossing angle is becoming popular in the design of electron–positron colliders. An example thereof is the SuperKEKB project, in which a collision with a large crossing angle is performed to boost the luminosity to0.8×1036cm−2s−1. Future circular colliders will also be designed with a large crossing angle. Strong-strong simulations, which have shown the first coherent head-tail instability, can limit the performance of proposed future colliders. The mechanism whereby this instability occurs is mode coupling due to the cross-wake force. This instability may affect all collider designs based on the crab waist scheme.