Conical intersections - phase
A rather quick googling shows up that there is some Longuet-Higgins theorem, which states almost what you are looking for. For instance, the abstract of this paper says:


  It is proved that if the wave function of a given electronic state
  changes sign when transported adiabatically round a loop in nuclear
  configuration space, then the state must become degenerate with
  another one at some point within the loop.


In the paper itself it is further stated that this in turn implies that 


  [the corresponding] potential energy surface intersects one of another electronic state


and that the theorem 


  makes it possible to diagnose the presence of an intersection between potential energy surfaces from the behaviour of the electronic wave function as it is transported adiabatically round a closed loop remote from the intersection


which answers both your questions, if I understand them correctly.