Why do strings split in string theory?

In string theory, we are told strings can split and merge if the string coupling is nonzero, even while the worldsheet action remains Nambu-Goto or Polyakov plus a topological term. However, a classical solution, in say the light cone gauge, shows that provided the worldsheet time increases with light cone cone time initially, adding topological terms will not change the solution, and the string will not split. So why and how do strings split?