"At loggerheads with someone" is usually used to describe a situation where two people or groups have conflicting opinions or ideas, and they can't seem to find a way to agree or compromise. If you're looking for other ways to express this idea, there are many synonyms to choose from depending on the context. For example, you could say they are in conflict, at odds, in disagreement, at each other's throats, having a standoff, experiencing friction or tension. Using these synonyms offers a better way to describe the situation without repeating the same phrase repeatedly. Ultimately, whichever synonym you choose, it should be able to convey that the relationship between the two parties is strained.