We study the asymptotic distribution of the eigenvalues of a two-by-two semiclassical system of coupled Schrödinger operators in the presence of two potential wells and with an energy-level crossing. We provide Bohr-Sommerfeld quantization condition for the eigenvalues of the system on any energy-interval above the crossing and give precise asymptotics in the semiclassical limit. In particular, in the symmetric case, the eigenvalue splitting occurs and we prove that the splitting is of polynomial order $h^{frac32}$ and that the main term in the asymptotics is governed by the area of the intersection of the two classically allowed domains. Our method consists essentially on two parts. A first part where we construct suitable $L^2$ solutions to the system in order to prove the existence of eigenvalues together with a rough estimate on their location. Then, a purely microlocal approach to get precise estimates. This is a joint work with Setsuro Fujiié (Ritsumeikan University, Kyoto).