At C2, in the English Grammar Profile, point 109 in the category of PRONOUNS and reciprocal is defined as:
‘each’ (+ noun or pronoun) as subject followed by ‘the other(s)’ as an object, to refer to two related things. (quantity)
Most of the examples in iWeb corpus seem related to law.
A looser search each * * * the other* *
1 EACH OTHER . ON THE OTHER 120
2 EACH OTHER . THE OTHER 39
3 EACH PARTY SHALL PROVIDE THE OTHER 21 This is folled by ‘the other party’ context
4 EACH TRYING TO OUTDO THE OTHER 21
5 EACH OF THEM PROVIDES THE OTHER 20
An interdependency relationship will be found to exist between two individuals if:
they have a close personal relationship;
one or each of them provides the other with financial support;
and one or each of them provides the other with domestic support and personal care.
6 EACH OTHER , AND THE OTHER 20
7 EACH PARTY REPRESENTS TO THE OTHER 20
8 EACH ONE DIFFERENT FROM THE OTHER 19
9 EACH . ON THE OTHER 17
10 EACH PARTY WILL NOTIFY THE OTHER 16
each * _v the other *
1 EACH SIDE ACCUSED THE OTHER OF 25
2 EACH SIDE ACCUSING THE OTHER OF 19
3 EACH SIDE ACCUSES THE OTHER OF 16
4 EACH SIDE BLAMING THE OTHER FOR 16
5 EACH SIDE BLAMED THE OTHER FOR 11
each * * _v the other
1 EACH PARTY SHALL PROVIDE THE OTHER 21
2 EACH TRYING TO OUTDO THE OTHER 21
3 EACH OF THEM PROVIDES THE OTHER 20
4 EACH PARTY WILL NOTIFY THE OTHER 16
5 EACH TEAM WILL PLAY THE OTHER 13
6 EACH PARTY SHALL NOTIFY THE OTHER 11
7 EACH SIDE HAS ACCUSED THE OTHER 9
8 EACH PARTY MAY DISCLOSE THE OTHER 8
9 EACH OF US THOUGHT THE OTHER 7
10 EACH PARTY SHALL INDEMNIFY THE OTHER 7