In the Co-Pt system, a simple cooling experiment can drive a sample ordered in the cubic L12 structure (Cu3Au type) close to the two-phase region involving L12 and the tetragonal L10 (CuAu type) structure. Using transmission electron microscopy observations, we show that the antiphase boundaries (APB’s) in the L12 structure are decorated by the L 1 0 structure and that the L 1 0 variant formed during this wetting process is related to the characteristics of the APB. The L10 tetragonal axis is normal to the displacement vector of the APB and the translational variant ensures the continuity of the platinum-rich cubic planes between the bulk and the wetting structure. To understand this peculiar wetting process, we develop different theoretical approaches based on a microscopic Ising model on the fcc lattice with interactions up to the second nearest neighbors. At 0 K, the model accounts for the observed selectivity of the wetting process. Then, using a mean field approach, our model predicts the wetting by the L10 structure at finite temperature, with a selectivity similar to that observed in the Co-Pt samples. Furthermore, the usual logarithmic divergence of the width of the wetting layer with respect to the excess free energy still holds. Finally, we use a general phenomenological Landau approach, where the symmetries of the fcc lattice and of the (vectorial) order parameter are taken into account, to show that the width of the wetting layer is very sensitive to the orientation of the APB. This phenomenological approach makes it clear also that the wetting of the APB in the L 1 2 structure by the L 1 0 phase, although observed here, is not unavoidable theoretically, which is not the case when the relevant order parameter is scalar.