Look, you're thinking of this in the extremes i.e. Nankangs vs R888s, when the difference between the compounds and treads will show up with ANY two tyre makes. I would no more put RE050s on the front and PS2s on the back as I would do the first combination mentioned. If you have tyres that give differing grip levels at different times and in different temperatures then at some point you ARE going to have an issue, and it's juts a case of how bad that issue is going to be. It might just be a an extra bit of slip at >5mph turning into your road, or it might be a full-on tank slapper into the armco on the motorway, and that's the problem: Unpredictability.
There isn't a car manufacturer in the entire world that puts mixed tyres on their cars, and there's a very good reason for that.