I'm afraid this might be an ill-posed problem, but I am not certain how to avoid it. Hopefully the answers can be helpful in getting enough knowledge to fixing that issue.
Let $W_1$ and $W_2$ be independent Wiener processes on some probability space with probability measure $P$.
To what is extent is it possible to construct an equivalent probability measure $Q$ such that $W_1$ and $W_2$ are correlated under $Q$?
How is this affected by relieving assumptions $W_1$ and $W_2$? - to for instance being Levy processes or continuous diffusions? That is I'm hypothesising that this possibility somehow depends on the richness of the process class and the option will somehow arise from increasing this richness.