Something acts like the surfactant to keep it from separating. Probably not mustard like in a salad dressing :).
If you are interested in this topic, I can recommend the book "Proof: The Science of Booze" [1] which has a chapter on yeast and a following one on fermenation.
Gum arabic or Xantham gum?
> Ingredients: Fresh Dairy Cream, Sugar, Alcohol, Maltodextrin, Milk Products, Cocoa extracts and flavours, Irish Whiskey, Colouring: 150b, Emulsifier: E471, Acidity regulator: E331
https://en.wikipedia.org/wiki/Mono-_and_diglycerides_of_fatt...
That being said, Bailey’s isn’t that shelf stable and will likely turn into undrinkable too faster than the average veggie oil will go rancid. And veggie oil can be stabilized to some extent by adding vitamin E.
Is the solution as simple as basing it on ultra pasteurized cream further preserved by the alcohol and sugar, all bonded forever by some mechanical homogenization process?