  We introduce a novel solution concept, denoted $\alpha$-dissipative
solutions, that provides a continuous interpolation between conservative and
dissipative solutions of the Cauchy problem for the two-component Camassa-Holm
system on the line with vanishing asymptotics. All the $\alpha$-dissipative
solutions are global weak solutions of the same equation in Eulerian
coordinates, yet they exhibit rather distinct behavior at wave breaking. The
solutions are constructed after a transformation into Lagrangian variables,
where the solution is carefully modified at wave breaking.
