Before revision:  We prove existence of solutions to the corresponding conditional McKean--Vlasov equation, by realising them as suitable limit points of the finite-dimensional particle system , and we show that the pathwise realisation of the common noise can both trigger and prevent blow-ups .
After revision:  We prove existence of solutions to the corresponding conditional McKean--Vlasov equation, by realising them as suitable limit points of a motivating particle system with common noise. Furthermore, we derive results on the occurrence of blow-ups, thereby showing how these events can be triggered or prevented by the pathwise realisations of the common noise can both trigger and prevent blow-ups .