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Locally maximally entangleable states (LMESs) constitute a large set of multipartite states, containing, for instance, all stabilizer states. LMESs are uniquely characterized by (2n−1) phases, where n denotes the number of qubits. We consider here those LMESs whose phases are either 0 or pi and present a multipartite entanglement purification protocol for arbitrary such states. In contrast to all previously known recurrence protocols, this protocol uses a different ingredient, which is required due to the quantum correlations contained in the various LMESs. We compare this scheme to previously known entanglement purification protocols and show that the direct purification performs better than previously known protocols.