In string theory, one wants to study the interaction of strings throughout time. The interacting strings sweep out a surface in Minkowski space. To write down an action for the string, we need to assume that the world sheet is at least differentiable (usually smooth). But if we assume that strings cannot move faster than light, we get an immediate contradiction. Because if the world sheet would be smooth, then at a spacetime point where strings join, the tangent plane would need to be space-like, which implies that the string moves faster than light.
It is obviously easier to handle string theory defined on smooth surfaces, but the above implies that this would violate fundamental principles in physics. So what is the solution?
(I know that this is not an issue if the metric of the target space is Euclidean. But from a physical point of view, we are interested in Lorentzian metrics. Also, a Wick rotation does not solve the problem, because in the end, the string would still need to move faster than light to sweep out a smooth surface.)