I live in the Northeast and we plant a bulb that sounds similar but it's an ornamental allium. It's grown for the purple flower heads (since the deer don't eat them) and the bulb sounds the same, it divides and the new bulbs also flower. It has just finished the bloom now. I wonder if it's the same thing that you're talking about?

Here is a link that might be useful: purple allium

It could be some kind of hard neck. Does the stalk that the bloom is on get hard, with the cloves in a circle around it? I have heard that softneck garlic can evolve into hardneck but I am not sure if that is true. Softneck garlic doesn't bloom. Garlic will spread and go a bit wild if left in the ground, I have found out first hand.

I do not believe this is an ornamental. I am sure it was grown to be used as garlic. The bulbs are pretty much like a bulb of garlic you buy in the store, although the individual cloves are larger and there are not as many cloves per bulb.
As far as hardneck or softneck, I do not know. This has a bulb in the ground with about a 4 foot stalk coming out of it with a tennis ball size purple blossom on the top. I pulled some up and the stem got a little dry and hard but it was laying in 95 degree sun in the back of my truck. I have read that hardneck garlic does not do well in the hot and humid south so I would imagine this is softneck.