When Oron took us to the Poleg Dunes nature reserve near Tel-Aviv, on the MGS trip to Israel in March, one of the interesting plants we saw there was a large-flowered onion,
Allium telavivense - a very local endemic, now rare in the wild, as the building of Tel-Aviv has now covered most of its former habitat:
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It is in cultivation and may occasionally be available from specialist bulb dealers.
A New Zealand test of 30 different onion species found that
A. telavivense - the only one of the 30 - seemed to be completely resistant to white rot, the bane of onion-growers (you can find an abstract of the scientific article
here). So Oron, maybe you should do a little judicious crossing with a large-bulbed eating onion! (I think considering its large flowers
A. telavivense itself has rather small bulbs, doesn't it?)