Suppose $X$ and $Y$ are two Brownian motions such that $|X|$ and $|Y|$ are independent. Then it is easy to show that $\langle X,Y \rangle =0$ using the Tanaka formula, for example, and thus $X$ and $Y$ are independent. However this method just applies to Brownian motion.

Now I have two nice processes $X$ and $Y$ with a nice state space (for example, two Brownian motions on a star (walsh brownian motions)) and I know that the associated reflecting Brownian motions (distance to zero) are independent.

Question: How can I deduce that $X$ and $Y$ are independent?

This seems to be true since independent reflecting Brownian motions dont vanish simultaneously after time 0. Could someone help me? Any proof in the case of Brownian motion without using the cross variation sould be a big step!