  We extend a model of positive feedback and contagion in large mean-field
systems, by introducing a common source of noise driven by Brownian motion.
Although the driving dynamics are continuous, the positive feedback effect can
lead to `blow-up' phenomena whereby solutions develop jump-discontinuities. Our
main results are twofold and concern the conditional McKean--Vlasov formulation
of the model. First and foremost, we show that there are global solutions to
this McKean--Vlasov problem, which can be realised as 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.
