I'd never heard of Tayberries until I found them recently at my local Farmer's Market. According to Wikipedia, the tayberry is a cross between a blackberry and a red raspberry. I found the flavor to be, surprisingly, a bit of both. I was skeptical that I would find both flavors, but I did. Note this: "Unfortunately, Tayberries do not pick easily by hand and cannot be machine harvested and so they have never become a widely grown berry crop." That explains why I'd never seen them before!
If you get a chance to try a tayberry, please do! I recommend eating them just as they are.