If you only consider the right-movers, the "fermonic string" and the "type II string" is the same thing! The word "fermionic" doesn't mean that it only has fermions; it means that it also has fermions. Taking only fermions from type II strings would have a wrong central charge i.e. conformal anomaly! Your difference of the two actions is completely incomprehensible to me; perhaps you thought that the fermionic string should only have fermions but that's not the case. And which imaginary action? In the Minkowski space, all actions are always real, they have to be real due to unitarity.