So what are the benifits between a miniphase and a iwata?
I think there are not really judgements made a priori about these, instead it is autotech's purpose to provide popular horns for those who wish to make use of them. There were some thought experiments and simulations that went behind many of these designs. Many of the ideas originated on DIYaudio's various threads on waveguides and horns (see JMLC on LeCleach, Geddes on Waveguides, manufacturing of waveguides, etc).
I am not 100% sure where the concept for minphase came from, but I have an idea. Anyone, I see it as too large for a given loading to be practical, but that is my $.02. It should have very good behaviour over its pass band, but a system is more than the sum of its parts, and the whole system needs to be taken into consideration when designing, not just a bunch of passbands.
The Iwata is a throw back to another idea and JMLC helped modernize its profile. It loads down low (the 300hz) while not being too large in the vertical, which really helps with integration. I've yet to see any real measurements for it.