The translation looks weird to me... "a vrea" is without wanting it, right?
I thought it would be "ei au adormit fara sa realiza".
Your reasoning is good. We would more likely say fără să realizeze / fără a realiza or, another way of saying that in Romanian, fără să își dea seama.
I was wrong and it suggested I use "au adormit fără să să realizeze!" as a correct answer, but that's two of "să" in a row? That doesn't seem right
It's a mistake.