Do magnetic fields exist beyond the gravity of a black hole?

The magnetic fields in question are generated by the motion of charged gas in the accretion disk i.e. outside the black hole. So it is not the case that the black hole generates the magnetic field which then has to somehow escape through the event horizon.

The literature contains references to the field lines threading the event horizon, but I suspect this is just rather casual wording. Real black holes do not contain a true horizon as the true horizon takes an infinite coordinate time to develop. At best the system will have an apparent horizon. It isn't clear to me whether magnetic fields can thread an apparent horizon, but in any case the field lines will become very distorted in the immediate vicinity of the horizon and the end result will be essentially identical to the field lines threading the horizon.

It has long been known that active galaxy nuclei possess a strong magnetic field, but what is new is the realisation of just how strong these fields are.