From Longman Dictionary of Contemporary Englishbe at each other’s throatsbe at each other’s throatsARGUEif two people are at each other’s throats, they are fighting or arguing → throat
Examples from the Corpus
be at each other’s throats• Congress and the President have been at each other's throats for so long that it's a wonder they can agree on anything.• Lisa and Nicole were at each other's throats the whole trip. 