I'm not sure they are in the real world yet. Maybe it's just a bigger theme park.
There's a fairly popular fan theory that this all takes place on the moon or Mars. Yet there's no evidence to indicate that this doesn't take place on the Earth.
Still, there are certainly some incongruities.
In season 1, MiB says the the real world is one of plenty, where people's basic needs are met and disease has been eradicated. And while people do have pretty decent living conditions, there is a wide gulf between rich and poor and poor people certainly do not live lives of plenty. And not only does disease still exist (exhibit A: Caleb's mom), Caleb struggles to pay for treatment. And there's also a pretty noticeable criminal element, with a disturbingly large amount of "wetwork" (murder)
It's also alluded to that there's some extensive eugenics going on - not just the elimination of genetic defects but also cultivating certain personality traits. Felix says something to Sylvester to that effect. This system apparently had one hell of an oversight with the MiB (and a small fraction of people who share his personality).
This isn't enough to call bullshit, and part of that can be explained by an unreliable pov, but the show in general seems to have all the hallmarks of on-the-fly-rewrites that leave these sorts of incongruities whenever the writers decide to take the show in a different direction.