Path integral quantization of bosonic string theory
In Minkowski spacetime, the action has to be real. Btw, that's necessary for the classical limit to give principle of least action. Yes, such sums are ill-defined, so some might say that the theory is mathematically defined by analytically continuing (Wick rotating) to Euclidean time, where you have nice exponentially decaying weights. You'll get saddle points given by the extrema of the action and you can expand around those solutions and deal with the theory perturbatively.
Think of the path integral in QM. Going from paths of least action (classical mechanics) to a weighted sum over all paths gives us quantum mechanics ("first quantization"). Going from quantum mechanics to QFT involves a sum over all field configurations ("second quantization"). Similarly, the moment you write out the path integral summing over all possible string configurations, you're studying a quantum mechanical system of strings.
Even in QFT when you do the Fadeev-Popov method and introduce ghosts, you have to quantize them so that diagrams with ghosts consistently cancel amplitudes of (some) diagrams with longitudinal gluons. I'm at a loss for a more insightful answer.
I am afraid, I am just going to provide the standard lore here. I will do so nonetheless as not many have attempted answering this question yet. Allow me look at the questions from simplistic QM(not QFT) picture. I'll take the question in reverse order: When did we quantize? Well, we never have to, that is the beauty(?) of path integrals! You just pretend your p's and q's are real numbers locating the particle in phase space. In absence of "momentum dependent potential", after the path integral program(PIP) all you are left with is sum (superposition) over histories giving you some amplitude. In QFT the extension of this naive picture only works for bosons and you have to introduce complications (grassmann nos for fermions; additional ghosts for gauge fields etc) so that PIP works (i.e. you have some classical field with given properties which is summed over all possible configuration). All this lets you bypass quantisation, deriving appropriate Wick's theorem and Feynman diagrams etc. Now, what if action S is real? Indeed it is real in usual QM, the lore is that only neighborhood of classical path contributes to the total amplitude which becomes superposition of "amplitude" contribution from individual paths. Consider a set of chosen path which is not close to the classical trajectory. Since for them phase factor is not stationary they could in principle have widely different values from one another. One expects that when you sum these phase factors there will be "wild" phase cancellations associated with it.