The non-stops are a seed-grown strain, so that's most likely how they were propagated. I carry some over every year, and I don't do much except keep them in a cool location during the hottest part of the summer (our heat and humidity is anathema to most tuberous begonias) and water them sparingly (I've lost more of them to overwatering than anything else). They get some osmocote in the potting mix at the beginning of the season, then maybe a foliar feed at some point, if I think about it, but I don't do much else with fertilizer, either. You have longer winters with which to contend than I do, so that might affect the length on storage time - you might have to start your tubers indoors next spring before putting them out, but you probably also have cooler night temps in the summer, so it might all balance out.

How do you store them?

Do they automatically form tubers? or should they be given more or less light to promote the tubers to grow?