<p>Although nonlocal elasticity theories have been intensively used to study various boundary value problems, they have not been proven to be well-posed at all for the general case. Therefore, it is extremely necessary to check the well-posedness of a boundary value problem of the nonlocal elasticity theories before solving it. In this paper, a well-posedness criterion for harmonic plane wave problems of Eringen’s two-phase nonlocal elasticity theory has been established. It says that a harmonic plane wave problem in domain with non-empty boundary of this theory is well-posed when none of its constitutive boundary conditions is identical to an equilibrium boundary condition of the problem; otherwise, it is ill-posed in the sense of infinitely many solutions. The proof of the criterion is based on the equivalent differential model of Eringen’s two-phase nonlocal elasticity theory. With this criterion it is easy to check whether a harmonic plane wave problem of the two-phase nonlocal elasticity is well-posed or not. As an example of application of the established criterion, the problem of SH waves propagating in traction-free two-phase nonlocal isotropic elastic half-spaces is considered. The well-posedness of the problem is checked using the well-posedness criterion. To find its solution we employ a novel method introduced recently (Vinh and Anh, in Proc R Soc Lond A 480(2293):20230814, 2024). Contrary to the Eringen method, the novel method satisfies the original equations of motion. Expressions of displacements, local and nonlocal stresses of SH waves have been found along with dispersion equation. The numerical results show that the nonlocality decreases the velocity of SH waves and the local parameter increases it.</p>