The only interpretation that has been worked out for all existing quantum theories is the one which says, observable X has value x with probability |<X=x|Psi>|^2. If you don't even have that, you have something less than a quantum theory; and if you do have that, you have a way to interpret it as making statements about some kind of reality. But yes, something is lacking; it doesn't tell you which observables are the ones that actually get to exist. You the user of quantum mechanics, are free to choose which observables you want to calculate.
The Everett interpretation proposes to obtain a more objective picture by saying, "the wavefunction" is what actually exists, and it contains multiple "worlds", only one of which is the world we observe. I should emphasize that in the only truly universal interpretation of quantum mechanics, the Copenhagen interpretation as originally understood, the wavefunction is not a physical object, it is just part of a calculation. In Copenhagen, the observables, not the wavefunctions, are what's real; even if later generations of physicists have got into the habit of regarding the wavefunction as a physical entity.
You suggest that the Everett idea of parallel worlds has been demonstrated for the nonrelativistic case, and can then painlessly be adapted to more advanced quantum theories, like relativistic quantum field theory. This is not true.
First of all, there is nothing like a consensus on exactly what worlds are hiding in a given wavefunction. You suggest that multiple worlds exist once there is decoherence. But decoherence is a matter of degree. Are you saying there is a particular amount of decoherence at which one world suddenly becomes many? If so, how much decoherence? If not, are you saying there is no objective fact about when one world becomes many? That wouldn't be very "realist".
Second, uplifting this to the relativistic case introduces extra problems. In relativistic QFT, wavefunctions are defined on spacelike surfaces. Are you going to say that a particular set of spacelike surfaces define an objectively preferred notion of simultaneity? That's against relativity.
There is a histories formalism (called decoherent histories or consistent histories) which is relativistic. But it doesn't ascribe reality to wavefunctions. Instead, like Copenhagen, it ascribes reality to observables. The observables can be attached to individual space-time points, which means that you don't need an objectively preferred reference frame.
But like Copenhagen, the histories formalism doesn't tell us which observables, at which space-time points, are the ones that take actual values. The only constraint is that the histories (each of which is a different ensemble of specific observables taking specific values) must all mutually decohere, as calculated by a so-called decoherence functional. An individual history could in principle contain just a single observable. People who want the histories formalism to provide a picture of reality, have suggested that the actual histories are as dense with observables as possible (as dense as is possible while preserving mutual decoherence), but while a reasonable criterion, this does not remotely pick out a unique set of histories as the objectively existing multiverse.
Then there's the problem of how to interpret probability in this framework. The decoherence functional also assigns an apriori probability to each history, and then the conditional probabilities of ordinary quantum mechanics can be derived from these. But what do these apriori probabilities associated with entire histories of the universe, actually mean? One way to interpret it would be, there was only ever going to be one universe, the specific universe was somehow selected at the start of time with that probability, and that's it.
But if you want an Everett-style multiverse interpretation of the histories formalism, then all the histories exist, but some of them have to count for more than others. The only way I can see to justify that, is to say that there are multiple copies. If the "probability of universe A" is twice that of universe B, there must be two copies of A and one copy of B. So you'd better hope that all your probabilities are rational numbers...
I've gone on about this at length, in order to convey some of the difficulties with the Everett concept, even in the form that I find most workable. And I haven't even got into the more technical problems of quantum gravity, where you can no longer take space-time points for granted, as an anchor for your observables.
Other "realist interpretations" have their own problems. Bohmian mechanics does not naturally become relativistic. Retrocausal theories like the transactional interpretation exist at a level of handwaving comparable to the many worlds interpretation. Objective collapse theories also have messy problems to solve once you want to make them special- or general-relativistic. There is no realist theory that has been worked out for all quantum theories.