Categorical Variable with Repeated Data Points


I have some data (continuous DV, nominal IV), that I would like to run an ANOVA or regression on. Strictly speaking, I have 3 levels, but there are some data points that belong to 2 or even all 3 levels (sorry, I don't know the statistical term to describe this). What is the best way to manage this? Do I count those points more than once (basically treat them independently for each category they belong to) or is there a better way? I've considered just creating a 4th "combined" level for these particular data points.

To top it all off, the data is non-normal and heteroscedastic, and if I add another level to the IV, the unequal sample sizes could be even worse. Any advice? TIA!
After more reading, it seems like I can just go on with dummy coding and enter "1" in whichever category applies, even if it involves multiple levels. And then I can proceed with linear regression? My data set has heteroscedasticity, even after transformation. Are there any relatively straightforward solutions to this? So far, I've been using Welch's ANOVA because of heterogeneity of variances, but I'm still not sure how to incorporate those data points that have multiple levels of the IV.