"-지만" is "but", conceding something before 먹다 "eat". The pattern is eating in spite of a drawback: I know there's a landmine in there but I'm eating it anyway, it's expensive but you've got to eat it, I can't cook but I can certainly eat, the first bite is bitter but it gets sweeter. It also runs the other way — I haven't tried it, but you try it and tell me; I bought it for the diet but never got round to eating it.