Kelsey, Did you notice the whitish stuff on the garlic before you planted? Or do you think it's in your soil? It may be limited to where you planted if it was on the garlic to begin with. Onions are vulnerable to it as well.
While they say there are no controls for white rot, it can't hurt to really, really ramp up
compost, as in layers several inches thick. I have a lot of faith in compost, and spraying soil with an EM mix. Don't let your soil get too damp, try to dry it out where you planted this batch, expose it to the sun, mix in compost, lots of it.
Permaculture prefers to plant things scattered around, and this seems to be a good example of why that is a good idea. If all the garlic is planted in one place then the problems can go from plant to plant. If pockets of garlic are planted in various places, some of it will make it where there isn't the sclerotium in the soil. Planting with a triad of herb/flower/vegetable helps with spreading out the crops, bringing in beneficial insects, fooling the bad insects.
The white rot might not be everywhere, so try some clean, clean new garlic in a place remote from where you planted it. Bag the bad stuff, and escort your white-rot garlic off the property!
Mediterranean climate, hugel trenches, fabulous clay soil high in nutrients, self-watering containers with hugel layers, keyhole composting with low hugel raised beds, thick Back to Eden Wood chips mulch (distinguished from Bark chips), using as many native plants as possible....all drought tolerant.
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