This is a follow-up question I have after reviewing this post: Difference in means statistical test for non-normal, heteroscedastic data?
To be clear, I am asking from a pragmatic perspective (not to suggest that theoretical responses are not welcome). When normality among groups is present (different from the title of the question referenced above), but the group variances are substantively different, what is the worst that a researcher might observe?
In my experience, the issue that arises the most with this scenario is "strange" patterns in the post hoc comparisons. (This has been observed both in my published work, but also in pedagogic settings...happy to provide details of this in the comments below.) What I have observed is something akin to this: You have three groups with $M_1 < M_2 < M_3$. The (omnibus) ANOVA gives $p<\alpha$, and the pairwise $t$-tests suggest $M_2$ is statistically significantly different from the other two groups...but $M_1$ and $M_3$ are not statistically significantly different. Part of my question is if this is what others have observed, but also, what other issues have you observed with comparable scenarios?
A quick review of my reference texts suggest ANOVA is rather robust to mild to moderate violations of the homoscedasticity assumption, and even more so with large sample sizes. However, these references do not specifically state (1) what could go wrong or (2) what might happen with a large number of groups.