Charged compact objects have been studied for the past few decades within the context of GR or any other modified gravity. So my question is : "What is the physical reason behind the existence of charge or electric field inside a neutron star?"
Bibhash Das The most obvious cases where an electric field is apparent is the case of a fast rotating neutron star. Here one can imagine a mechanism whereby neutrons close to the axis of rotation decay into protons and electrons. Then the protons are likely to remain bound to the rotating star and form a moving electric charge. The resulting magnetic field would then eject the electrons in a jet projected along the axis.
A similar effect is observed in jets from rotating black holes which suggests that black holes contain a neutron star which is the cause of the event horizon:
The same could apply to almost any astronomical object, like earth or planet. Overall electrical neutrality, as in ordinary matter. That is why gravity dominates, instead of much larger electrical forces.
To talk about actual magnetic monopoles is risky buisness...