I also find Tegmark's (and to an extent Schmidhuber's) multiverse theories attractive, I'm quite leaning towards believing they must be true.

But, if I'm not mistaken, the multiverse hypothesized by Hawkin, and the usual multiverse from String theory is not this mathematical multiverse. In String theory multiverses, not every possible universe is allowed to exist, only universes satisfying string theory conditions. That's not to say those hypothesis are inconsistent -- the String thoery multiverse may exist within the "Mathematical multiverse". Once we accept that falsifiability isn't the only motivation to create theories (say by subscribing to MUH), then we need to accept that maybe our universe is also part of a larger non-observable universe, if and only if this provides a particularly "simple" explanation to our existence.