- the tube's axis of rotation will appear to move relative to the walls as the asteroid
- you will have to use some complicated form of "compound" rotation
- you will align the axis for rotation of all tubes parallel to the asteroid's axis of rotation
Actually, an ability to precess was already designed into some O'Neil colony designs. With 2 artificial gravity tubes near to each other, they can push and pull so that they change the direction they face. This was proposed as a way to keep facing the sun, which obvious means they turn a full circle roughly every year (depends on the orbital position). For the asteroid rotation, we could be talking about rotation rates of 8 hours or less. Whether the forces involved and the unwanted acceleration on the surface are small enough is a question that is still unresolved.
This is a type of topic which I might return to sometime, but I've decide that I'm going to do a few micro-posts from the long list of things in my backlog where I introduce the subject in a few paragraphs.