Independence in randomizations

The randomization of a complete first order theory $T$ is the complete continuous theory $T^R$ with two sorts, a sort for random elements of models of $T$, and a sort for events in an underlying atomless probability space. We study independence relations and related ternary relations on the randomization of $T$. We show that if $T$ has the exchange property and $\mathrm{acl}=\mathrm{dcl}$, then $T^R$ has a strict independence relation in the home sort, and hence is real rosy. In particular, if $T$ is o-minimal, then $T^R$ is real rosy.