How "real" are the worlds of the many-worlds interpretation? Are there different interpretation variants of the MWI with varying conceptions of how real these worlds are? How "real" are those many worlds according to the interpretation/s?
Is it better thought of as different physical universes or one universe with different history contents with those histories maybe even somehow being entangled with each other or e.g. being as real as our current reality but only for some short time or being real only in the way in which they effect actualized reality or being real but "empty worlds"?
According to Wikipedia:Debate whether the other worlds are real  there is a debate about whether the other worlds are real. How (and why) would MUWIs describe the other worlds in terms of how "real" they are/what they are?
 A: Many worlds is a misnomer. It just refers to the fact that the universal wave function contains distinct terms linearly summed together, just like any quantum state. It is no more and no less than this. This interpretation should be called an Everettian or orthodox quantum mechanics interpretation. There are no sci fi alternate realities that one can jump between or something. The latter is the imagery that is conjured for me when I hear many worlds, and it’s just not a good description of the mathematics in my opinion.
Edit: Regarding the distinction between "real" or "unreal" worlds... This is some of that crappy philosophical speculation** mentioned below in the comments. If you adhere to the Everettian interpretation then you believe that the state of the universe is described by the universal wavefunction, full stop. In this case any term of the wave function is as real (or unreal) as any other. My gut feeling is that in the debate about whether the "other worlds" are real or unreal, the idea is that the world we experience is "real" while other worlds are "unreal". I would argue this is in direct contrast to the spirit of the actual Everett interpretation. Perhaps the bolded sentence answers the question...
**Note that, unlike many physicists, I do not deny the possibility of very good philosophical speculation about interpretations of quantum mechanics and the Everett interpretation. This is just not it.

edit to address many comments:
First two references since they were requested. I'll give more details about the references in my responses below.

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*Schlosshauer M, "Decoherence and the Quantum to Classical Transition"

*Maudlin T, "Three Measurement Problems"

*Barrett J, "The Quantum Mechanics of Minds and Worlds"
Now specific responses.

@Andrew: I always felt MWI does account for how our consciousness perceives one state of the world -- different "realizations" of our consciousness live in distinct branches of the wavefunction (and because of decoherence these different branches will not interfere). To me it seems what MWI doesn't do is explain the origin of the Born rule, which is somewhat at odds with the premise of unitary evolution since it assigns probabilities to distinct outcomes.

No, MWI does not account for our conscious perception of the world. This is the whole problem with MWI interpretation. But I thank you for your text here because it concisely captures one of the biggest misconceptions about the Everett interpretation. You mention there are distinct branches of the wavefunction. On this we agree, I would call them different terms in the universal wavefunction. But then you make the fatal error of relying on "decoherence" to ensure these branches never interfere. The problem is that while, in practice, decoherence avoids interference between these different terms, in principle, the branches are still able to interfere. For example we could put a human in a sealed ball, run that ball through a giant two slit experiment, and then re-interfere the human with themselves at the other side. (Credit for this thought experiment to Jess Reidel. It is similar to a Wigner's Friend experiment). If the results of such an experiment are what we expect from orthodox quantum mechanics, then we can not rely on decoherence as a crutch to allow us to only consider mental states along singular branches. That is,  in a true Everettian interpretation we MUST face the fact that the human brain can be in a meaningful (read: can be interfered) superposition state. I'll just add the final statement that there is no secret sauce to decoherence. Decoherence is there in regular quantum mechanics. At a pure wavefunction level (remember the universe has one wavefunction so we need a description that doesn't rely on tracing over degrees of freedom to get a density matrix), decoherence is just entanglement with the environment.
About the Born rule, I would reiterate that the deeper problem with the Everett interpretation is that it does not give us any recipe to relate the physical description of the world (universal wavefunction) with our subjective experience. In Copenhagen interpretation this is done as follows: 1) waveform evolves unitarily -> 2) wavefunction collapses according to Born rule -> 3) human body and brain is in a "well-defined" state (at least of certain observables) so we assume there is a 1:1 correspondence with a 4) well-defined mental state. So yes, we have the Born rule and it helps us extract mental states from physical states, but the real help in this interpretation is that it gives us a well defined physical state, just like we had in classical mechanics, to which we can attach a mental state without much fuss. So I would say, it is not the Born rule that Everett interpretation is missing, but rather ANY way to correlate mental and physical states. I would say attempts to "derive the Born rule" are fundamentally misguided and are red herrings. I would say we don't need to do any physics to complete the Everett interpretation, only philosophy. The physics is clear: unitary evolution under a Hamiltonian. There's not much more can be done there.

@bolbteppa Can you add some more detail to this, you make it sound very simple, but then one goes and looks at other sources and it's just a word salad which does not convey the simplicity you are indicating is there. A reference making it sound this simple would be great too.

The math of the Everett interpretation is as simple as can be for quantum mechanics. That's one of the great features about it. Sources on it and word salad that does not convey simplicity. This is exactly the philosophical garbage that I'm talking about. You can start with the references I mentioned above. The Schlosshauer book on decoherence is a great primer to different interpretations of quantum mechanics. It is also a great explanation of decoherence, and it is also great at explaining why decoherence does NOT solve the measurement problem, as many erroneously claim it does. If you google the book you will find an arXiv article which has similar content as the book that you can read for free. The second reference I give is the best explication I've found of the measurement problem (if anyone knows of a non-pay walled version of this article that would be really helpful). It says that there are 3 things that make having a good interpretation of quantum mechanics difficult. 1) The desire for unitary evolution of the wavefunction, 2) The desire to have local dynamics (i.e. no non-local hidden variables) and 3) The desire for measurements to actually have well-defined and "singular" outcomes. The paper explains how all interpretations we know of today reject at least one of these 3 points. For example Copenhagen and spontaneous collapse models reject the first, De Broglie Bohm mechanics rejects the second, and Everett interpretation and its siblings reject the third. For me it gave me a firm basis on which to think about interpretations of quantum mechanics.

@Filip Milovanović But you see, here again we run into interpretational problems, not of quantum mechanics, but of what people mean in their informal characterizations. For example, you say "There are no sci fi alternate realities that one can jump between or something", but if you could jump between them, then that wouldn't be "many worlds" for many people, just a single world with interconnected regions. I think that what the OP is asking is more along the lines of are these "other worlds" abstract mathematical entities that make the math work out, or are they physical. (Which then comes down to interpretation of what the Everettian formulation implies.)

Ummm... yeah this is why I hate the name "Many Worlds Interpretation". It's just not clear how what you're talking about relates to the unitary evolution of the wavefunction... The way you talk about many worlds makes sense if it were the plot of a sci-fi film, but it doesn't make sense in the context of quantum mechanics so it's hard to even address the question. But to try to address your last sentence "I think that what the OP is asking is more along the lines of are these "other worlds" abstract mathematical entities that make the math work out, or are they physical." On the Everett interpretation all of the terms of the wavefunction are equally real. On the Everett interpretation there is no preferred term of the wavefunction. And it would be strange if there were a preferred term because this term could, in principle, be split and recombined with another one like I described above in the human-in-a-ball interference problem. It's hard to imagine physical ground on which you could "prefer" some branch. I know there is a lot of research of people trying to figure it out. My personal feeling is that orthodox quantum mechanics doesn't leave many "handles" for us to find physical grounds to ever prefer one term of the wavefunction over another. Again, I think the burden of solving the measurement problem, if we commit to an Everett interpretation, falls to philosophers of the mind rather than physicists.

@mYnDstrEAm You didn't really answer the question. You basically said the MWI is just maths (and not so much an interpretation) then with later edits you said that in the MWI those other worlds are as real as ours. Besides that you seem to partly contradict yourself or at least don't make things clear, I didn't (just) ask about a "degree" of realness but about in what sense they are real...different physical universes just as physical, actual and real as ours? Histories not experienced by anybody within our universe? Empty universes without matter? etc...

I hope my long explications above help answer the question. But I think my point is that the question is of the not even wrong kind, or that it makes a category  mistake. The point is the different terms of the wavefunction don't really behave like what we might think of as worlds. They can be in superpositions, the nature of the superpositions change if you use a different basis, they can interfere with eachother, they can be entangled. All of these things are outside of what I personally would think of when I use the term "worlds". I wouldn't say the different terms of the universal wavefunction represent different physical universes. Rather, on an orthodox Everett interpretation, there is one universe and it can have a universal wavefunction which is a horribly complicated entangled superposition of the different physical particles and fields that we know of. Every term is just as real as every other term. "Histories not experience by anybody within our universe": this is a dicey question, but a good one. As I've said multiple times above: The issue with the Everett interpretation is that it doesn't leave us with any way to make sense of our subjective experience in light of the form of the universal wavefunction. There are some ideas for how this might work, but most are unsatisfying. See The Quantum Mechanics of Minds and Worlds and other works by Jeffery Barrett on the Everett interpretation. He is a great scholar of Everett interpretations and the measurement problem in general. I really admire his work. He gives a number of possibilities of how we might relate physical and mental states in Everettian and other interpetations. "Empty Universes without matter..." No. All of the terms of the wavefunction will have the same mass energy, and the same values for any other conserved quantities. Some people wonder how mass is conserved when worlds split. Does the mass double then? No... such statements betray a massive misunderstanding of what the Everett interpretation is. And I get so frustrated because this misunderstanding is fueled by the name "Many Worlds Interpretation". When an electron is split in a double slit experiment do we get two electrons for a total of twice the mass? No, we have on electron in a superposition of two different physical states. So it is with the different terms of the universal wavefunction.
