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Spin density waves, based on modulated local moments, are usually associated with metallic materials, but have recently been reported in insulators which display coupled magnetic and structural order parameters. We discuss one such example, the multiferroic Cu3Nb2O8, which is reported to undergo two magnetic phase transitions, first to an unknown antiferromagnetic phase at TN≈26.5K, and then to a helicoidal structure coupled to an electric polarization below T2≈24K [R. D. Johnson et al., Phys. Rev. Lett. 107, 137205 (2011)] which breaks the crystallographic inversion symmetry. By analogy with other complex oxides, one might naturally expect this intermediate phase to be a spin density wave phase. We apply spherical polarimetry to confirm the low-temperature magnetic structure, yet only observe a single magnetic phase transition to helicoidal order. We argue that the reported unknown phase actually supports an imitation spin density wave which originates from a decoupling of the components of the magnetic order parameter, as allowed by symmetry and driven by thermal fluctuations. This provides a mechanism for the magnetic, but not nuclear, structure to break inversion symmetry thereby creating an intermediate phase in the proximity of TN which imitates a spin density wave. As the temperature is reduced, this intermediate structure destabilizes the crystal such that a structural chirality is induced, as reflected by the emergence of the electric polarization, and the imitation spin density wave relaxes into a generic helicoid. This scenario in which critical fluctuations allow the magnetic structure to break inversion symmetry while the crystal structure remains centrosymmetric might be relevant to other complex multiferroics.