We just had our first garlic harvest this year. It's so easy to grow on your own, even in small places, just plant in the fall and mulch it.
The crop of garlic here was harvested in July and then just dried until now. I cleaned them up and see no rot, thank goodness. Maybe you are just getting last year's crop??
I read a couple articles, with nice pictures. Maybe you can identify the problem comparing the garlic you have with these two sites pictures? Hope this helps.
Garlic: Organic Production
Mites, fungus, over watering, poor storage, could be anything.