An example of what you get when you apply what you think is a rule in defiance of what sounds natural, from www.cbc.ca :
The Green party and the Bloc Québécois each has nine per cent.
The mistake is in thinking that it has to be “has” because of the word “each.” But the word “each” is not the grammatical subject of the sentence. The two party names are, as a coordinated compound subject. So of course it’s “have.” “Each” is an adverb, and could have been moved to the end – which would have made the grammatical structure more obvious:
The Green Party and the Bloc Québécois have nine per cent each.