  We show existence of a global weak dissipative solution of the Cauchy problem
for the two-component Camassa-Holm (2CH) system on the line with nonvanishing
and distinct spatial asymptotics. The influence from the second component in
the 2CH system on the regularity of the solution, and, in particular, the
consequences for wave breaking, is discussed. Furthermore, the interplay
between dissipative and conservative solutions is treated.
