Technically, that "core" is actually the stem.
Most authorities do recommend that you remove it once it turns green, because that would make the garlic bitter. And, years ago, I dutifully did that. Turns out, it was one of those conventional wisdom things that everybody believed, but nobody reality tested.
What got me going was the uprise of green garlic as a commodity. Green garlic is merely scallions grown from garlic instead of onions. "Hold on," I reasoned. "If the growing plant is so good, how could the stem not be?" So, I started experimenting. Lo and behold, I have never noticed any bitterness from leaving the stem in place.
As FrenchFries notes, once it turns green it doesn't have the same garlicy flavor as the rest of the clove. And it can be tough. But, overall, leaving it in will not affect a dish pro or con.
Something else I've noticed is that none of the celebrity chefs trim away the root end of a garlic clove. That's one I don't understand, because that part is woody and tasteless, as well as being ugly. I always trim it away.