Depends upon what you mean by realism. If you just mean belief in a physical reality independent of a conscious observer, I am not really of the opinion you need MWI to have a philosophically realist perspective.

For some reason, everyone intuitively accepts the relativity of time and space in special relativity as an ontological feature of the world, but when it comes to the relativity of the quantum state, people’s brains explode and they start treating it like it has to do with “consciousness” or “subjectivity” or something and that if you accept it then you’re somehow denying the existence of objective reality. I have seen this kind of mentality throughout the literature and it has never made sense to me.

Even Eugene Wigner did this, when he proposed the “Wigner’s friend” thought experiment, he points out how two different observers can come to describe the same system differently, and then concludes that proves quantum mechanics is deeply connected to “consciousness.” But we have known that two observers can describe the same system differently since Galileo first introduced the concept of relativity back in 1632. There is no reason to take it as having anything to do with consciousness or subjectivity or anything like that.

(You can also treat the wavefunction nomologically as well, and then the nomological behavior you’d expect from particles would be relative, but the ontological-nomological distinction is maybe getting too much into the weeds of philosophy here.)

I am partial to the way the physicist Francois-Igor Pris puts it. Reality exists as independently of the conscious observer, but not independently from context. You have to specify the context in which you are making an ontological claim for it to have physical meaning. This context can be that of the perspective of a conscious observer, but nothing about the observer is intrinsic here, what is intrinsic is the context, and that is just one of many possible contexts an ontological claim can be made. Two observers can describe the same train to be traveling at different velocities, not because they are conscious observers, but because they are describing the same train from different contexts.

The philosopher Jocelyn Benoist and the physicist Francois-Igor Pris have argued that the natural world does have a kind of an inherent observer-observed divide but that these terms are misleading being “subject” tends to imply a human subject and “observer” tends to imply a conscious observer, and that a lot of the confusion is cleared up once you figure out how to describe this divide in a more neutral, non-anthropomorphic way, which they settle on talking about the “reality” and the “context.” The reality of the velocity of the train will be different in different contexts. You don’t have to invoke “observer-dependence” to describe relativity. Hence, you can indeed describe quantum theory as a theory of physical reality independent of the observer.