Might be sand in the paint? You can buy washed sand at the paint store. Mix it with paint in a separate container from the original can and keep track of your proportions so you can reproduce the texture you like. The make a special roller cover for applying sand finishes.
I don't think you can put a lime finish over existing latex paint but it has been awhile since I thought about it.
I am assuming you are trying to reproduce a paint finish of course. The other post makes more sense if you are doing new walls on the lathe.