> OK, I guess you need Born's Rule, too. But so does every other interpretation, including the Copenhagen one.

Sure (note that some people claim that Born's rule can be derived from other considerations but the issue remains highly controversial). And at what point do you need Born's Rule? (hint: m9t)

I've read enough about the subject to know that there is not "one" MWI but many different variants (that's of course also the case for the Copenhagen Interpretation, which is even less well-defined). DeWitt (who introduced the "many-worlds" label, by the way) seemed to believe that branches are more "real" than Everet did, for example. And I don't know where do you put the "consistent histories" formulation (some people say it's many-worlds, some people say it's neo-Copenhagen).

Anyway, you have not yet explained to me how do you get predictions about experimental outcomes.

This is what happens in standard QM:

1) You have a wave function describing the system of interest (psi), we assume it was prepared in that state.

2) You can calculate the possible outocomes of a measurement, say +1 with probability 50% and -1 with probability 50%.

3) You perform the experiment, the outcome is +1

4) Immediately after the measurment you describe the system with the wave function psi+ that is the eigentstate corresponding to the outcome +1 (for -1 it would be psi-).

5) You can use the wave function psi+ to make predictions for new experiments, for example repeating the measurement before the state has time to evolve (according to Schroedinger's equation) you will get +1 again.

If you think the MWI gives the same predictions, this is my best guess of how you may think it works:

0) There is a universal wave function describing the whole UNIVERSE (PSI)

1b) Some mathemagical operations can be used to extract a wave function that describes the physical universe (the branch of the UNIVERSE that we are in, however you want to call "our world" out of "many worlds"). Let's say we decompose it as psi for the system (which in "our world" has been prepared in that state, so it's described as in the standard formulation) and psi' for the rest of our universe (not the UNIVERSE, mind you).

2) You can calculate the possible outocomes of a measurement, say +1 with probability 50% and -1 with probability 50%.

3b) You perform the experiment, but you don't really do perform an experiment because the only thing that happens is that the UNIVERSE evolves unitarily and PSI follows Schroedinger's equation in a completely deterministic fashion. However, somehow you do get a random outcome. You do also get the other outcome, but that's in another branch (or whatever). In your branch, you get an outcome with is either +1 or -1 as predicted. Remarkably, if you repeat the experiment the frequency of the different outcomes will match the predicted probabilities (unless you're unlucky, because there are other branches where you don't get the expected frequencies and your predictions will fail). Anyway, say that in your branch you got the result +1.

4b) You do the mathemagical operations of step (1b) again, but in this case to get the wave function describing your branch you don't say "in my world I have a system prepared as such and such", you say "in my world I had a system prepared as such and such and then I did a measurement and I got +1 as outcome". This gives you the wave function for your universe, that can be factored as psi+ (because you got the result +1) for the system of interest and psi'' for the rest of our universe.

5) You can use the wave function psi+ to make predictions for new experiments, for example repeating the measurement before the state has time to evolve (according to Schroedinger's equation) you will get +1 again.

As far as I can imagine, to make predictions about "your universe" you have to include the outcomes of the previous measurements to get the right description for the system (i.e. you need to recover the "collapsed" wave function psi, even though PSI doesn't collapse). That's what I mean by putting the collapse back in somehow.