"is the physical world actually real?"- The words physical and real are very close together in my map and with out know how they are spaced in your map I can not gain traction in helping answer this question for you.

"In what sense does the world exist?" In the sense that it is repeatable. I currently live under the assumption that this repeatability derives from the world being external to myself and objective; this map as of yet fits what I have observed of the territory.