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.