each has/have

Which is it used in these examples ,the singular or the plural form?

e.g. The traveler and the local each has/have reason to love the beach.

Each traveler and each local has/have reason to love the beach.

The traveler and the local each has reason to love the beach.

The traveler and the local each have reason to love the beach

OK, according to the Scott, Forseman Handbook for Writers, the verb in these can be either love or loves. It’s only when each is before the compound subject that the verb has to be the singular loves.

Each traveler and local has reason to love the beach.