A basis of identities for an algebra is irredundant if each of its proper subsets fails to be a basis. A sufficient condition is established under which a non-finitely based finite algebra of finite type has no irredundant bases. This result is then used to construct the first known trio of finite involution semigroups, all sharing a common semigroup reduct, such that one has a finite basis, one has an infinite irredundant basis, and one has no irredundant bases.