In the Co-Pt system, a simple cooling experiment can drive a sample ordered in the tetragonal L10 structure (CuAu type) close to the two-phase region involving L10 and the cubic L12 (Cu3Au type) structure. Using transmission electron microscopy observations, we show that interfaces in the L10 structure are decorated: orientational domain walls are wetted by a single layer of L12 structure whereas three macroscopic layers (L12/L10/L12) appear at the antiphase boundaries. We then analyze this complex behavior in the framework of the Ising model with interactions limited to first and second nearest neighbors. This approach is generic in the sense that it is the simplest one that reproduces the L10 and L12 ground states, without the specificities of the model with first nearest-neighbor interactions only. The finite-temperature properties of the various L10/L10 interfaces are computed with a low-temperature expansion and cluster variation method calculations in the inhomogeneous tetrahedron-octahedron approximation. The results are in full agreement with our experimental observations concerning the wetting of interfaces.