On a projective complex surface, a curve $C$ may sit in a linear equivalence class $[C]$ that is given as a sum $[C] = [C_1] + [C_2]$, where $C_1$ and $C_2$ are effective. In general the curves in the class $[C]$ are not necessarily all reducible, though they may be: for example, there are both reducible and irreducible curves in the class $[H]+[H]$ on $\mathbb{CP}^2$ where $[H]$ is the hyperplane class.
Now further suppose that $[C_1]^2 < 0$ (EDIT: and that $C_1$ is irreducible), implying that $C_1$ is the only curve in its linear equivalence class. Is it now the case that every curve in the class $[C] = [C_1] + [C_2]$ is reducible with one component given by $C_1$? Or, if not, are there further restrictions on $[C_1]$ and $[C_2]$ that do force this to be true?