I am not familiar with this plant but couldn't resist trying some of the little bulbs. One now has new growth. The other one in this pot is still firm but no growth yet. The other little pot has three firm bulbs in it with no new growth on any of those.
Mine likes water and bright light. It blooms in the late summer/early fall with a long flower stalk for many small white stripped flowers. It makes lots of bulblets and they fill the pot over time.
I was mistaking this for Ornithogalum longibracteatum, or pregnant onion. The B. volubilis (sea onion) is nice too but has little tiny flowers on their long climbing stems. Mine is flowering now. I don't stake it per se but I fine it something to climb up on. Same thing I guess. When bulb reaches some critical point it splits in to two. It grows more in the winter and likes plenty of water then.