From a theoretical perspective, nothing, it cannot happen. growing to adolescence in a suburb of the fictional extinct planet Krypton, she is immune to all Earth germs. I never recall her sneezing in the Comic books. She would be the perfect image of health. She could be vulnerable to diseases from her former world- and such things as kryptonite and Red suns, under which Superman"s ( her also) Powers are ineffective. when she visted Kandor, she did not have Super powers.
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Appa sneezes on him
Black Supergirl
SuperGirl saves the world like SuperMan.
Supergirl - Matrix - was created in 1988.
Superman Supergirl was created in 1998.