  Quantum nonlocality is usually associated with entangled states by their
violations of Bell-type inequalities. However, even unentangled systems, whose
parts may have been prepared separately, can show nonlocal properties. In
particular, a set of product states is said to exhibit "quantum nonlocality
without entanglement" if the states are locally indistinguishable, i.e. it is
not possible to optimally distinguish the states by any sequence of local
operations and classical communication. Here, we present a stronger
manifestation of this kind of nonlocality in multiparty systems through the
notion of local irreducibility. A set of multiparty orthogonal quantum states
is defined to be locally irreducible if it is not possible to locally eliminate
one or more states from the set while preserving orthogonality of the
postmeasurement states. Such a set, by definition, is locally
indistinguishable, but we show that the converse doesn't always hold. We
provide the first examples of orthogonal product bases on
$\mathbb{C}^{d}\otimes\mathbb{C}^{d}\otimes\mathbb{C}^{d}$ for $d=3,4$ that are
locally irreducible in all bipartitions, where the construction for $d=3$
achieves the minimum dimension necessary for such product states to exist. The
existence of such product bases implies that local implementation of a
multiparty separable measurement may require entangled resources across all
bipartitions.
