"Type III error occurs when you correctly conclude that the two groups are statistically different, but you are wrong about the direction of the difference. Say that a treatment increases some variable. But in your experiment, random sampling leads the value of that variable to be lower (on average) in the treated group, and enough lower that the difference is statistically significant."
It sounds like a test which is better able to find significant differences (i.e., more powerful) should thus mechanically lead to more type III errors. Is there any literature on the topic?