So you call it aioli instead of alioli? Funny I thought it was a typo. The word comes from all (ajo, garlic) and oli (aceite, oil) in the Catalan speaking side of Spain.
The authentic is made of garlic, virgin olive oil and a tiny bit of salt. No eggs required, but you may use one if the result using just oil and garlic is too strong, so it comes out as a heavy garlic scented mayonnaise. It has to be hand made in a mortar, with a lot of patience and adding the oil very slowly, a couple of drops at a time. You first smash 3 or 4 fresh garlic cloves until they become a paste that could be spread on a toast, then you go adding the oil very slowly. If you try to speed it up it will "cut" (I don't know and can't find the word in English) and the sauce will be spoilt forming unmixable bubbles. There are ways to recover it, but it's not easy. That's the way I used to make it, not anymore for it's too strong for our aging stomachs. Once it's "locked" you may add a few drops of Tabasco or any other spicy sauce you wish. We don't do that, some people add just a few drops of lemon juice. I guess you can add bacon grease or trying to make the whole thing with liquid bacon grease, although probably it will become solid like butter once done.