So you've got two modes of reproduction with Allium. Allium like this typically follows a biennial habit, so this years garlic will split into cloves around the fall, in preparation for sending up a flowering stalk next spring/ summer. The cloves are vegetative propagules; just another way to get more garlic other than seeds. Hence you can just plant a clove and get a garlic next year, or, you can plant seed and also get garlic.
Now for your actually question, I believe the segmentation is probably exogenous, technically yes, however, I am by no means an expert in Allium morphology (although I have done graduate coarse work in plant morph, and worked in a plant morph lab), so don't quote me. However, it wouldn't appear like you are describing. Think of the ring at the base of a clove of garlic as a bunch of 'stems'. The branching would originate there.
I've enjoyed your wisdom so much lately! Thank you for sharing it!!! I'm learning about plant propagation in general... Is that ring at the base of a clove the same thing as a rhizome?
This is amazing info to me. I've been growing garlic at a hobby level for ages and never knew how the bulbs develop. Thank you for sending me down a garlic education rabbit hole!