First of all, scientists would scramble trying to study the portals. These portals would help us understand space and time or some stuff I don't know what the scientists will find out. The next is military interventions and quarantines as the scientists would inadvertently piss off some monster, making it charge at the portals with the need to murder everyone, this will catch attention of the Hunter's Guild. The other thing that would also happen while military interventions are happening is interdimensional(?) diplomacy as the people of Earth and Monhun!Earth interact, especially through the portals at the 5 cities that you've mentioned.
Now after that happens, we can either try to economically colonize Monhun!Earth, take them over by force, or to seal the portals. Whichever happens, we will piss off just about everyone.