We prove that if $T$ is a complete theory with weak elimination of imaginaries, then there is an explicit bijection between strict independence relations for $T$ and strict independence relations for $T^{ \text{ eq}}$. We use this observation to show that if $T$ is the theory of the Fra\"{ i}ss\'{ e} limit of finite metric spaces with integer distances, then $T^{ \text{ eq}}$ has more than one strict independence relation. This answers a question of Adler [1, Question 1.7].
Categories: math.LO