I ship it
to 
"Northman. There are motions I would like to show you."
"I know you are strong. I wonder if you have stamina."
"Finally, a man who won't break."
http://files.gamebanana.com/img/ico/sprays/54cd7877cbf11.gif
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"Sometimes, shield becomes smashing board." "Stand behind Braum!"
Lore wise, I wouldn't be surprised that Illaoi likes Braum, but I don't think he would have anything to do with her.
Braum lives by a moral code, certain things are right and others wrong. And he strives to be good and kind by helping, supporting, and protecting people who live by a similar moral code from those who don't, particularly the weak and innocent. He actively strives to protect and support the world around him and make it a better place.
Illaoi lives by survival of the fittest, doesn't even believe in good and evil, and thinks that living 'right' is sating ones own desires and imposing ones own will pitted against the will of those around you. Life is a struggle that pits every person against every other and that struggle is to be enjoyed and celebrated regardless of the outcome.
They're complete opposites. And while Illaoi might be fine being with someone like Braum, because she doesn't believe anything is 'wrong' so long as you are living your life, Braum does not share that belief, and is such a morally driven and emphatic person that he wouldn't be okay being with someone like her, and would likely stand against her imposing her will were they ever to meet.
"Stay away from her... or my ship, landlubber!"
Tbh, I rather assumed 
was already a thing.
Needs a return address 'cause this ship is going back.
No one can get inbetween Illaoi X gangplank. Or My Lissandra X Braum