So I have this problem as well and also am using an ATI R9 200 series graphics card, along with a 6 core i7 processor. After some extensive troubleshooting, I discovered that the program seems to have troubles running on more than 2 cores. So here is the solution:
Make sure to turn off all compatibility settings for white.exe that you may have previously done.
Then create a new shortcut and enter this as the location (change white.exe location if you've installed to a different directory):
C:\Windows\System32\cmd.exe /c start "White" /affinity 3 "C:\Program Files (x86)\Lionhead Studios\Black & White 2\white.exe"
After you've created the shortcut, right click on it and choose Properties. Make sure that the "Start in" field says the following (or whatever location you installed BW2):
"C:\Program Files (x86)\Lionhead Studios\Black & White 2"
You can also change the icon to the white.exe location as well. This fix forces white.exe to run with 2 cores and will fix the texture issues you have popping in and out as well as most other anomalies. Let me know how this works for you!