In 1985 Kaspi provided a single-letter characterization of the
sum-rate-distortion function of a two-way lossy source coding problem
in which two terminals send multiple messages back and forth with the
goal of reproducing each other's sources. Yet, the question remained
whether more messages can strictly improve the sum-rate-distortion
function. Viewing the sum-rate as a functional of the distortions and
the joint source distribution and leveraging its convex-geometric
properties, we construct an example which shows that two messages can
strictly improve the one-message (Wyner-Ziv) rate-distortion
function. The example also shows that the ratio of the one-message
rate to the two-message sum-rate can be arbitrarily large and
simultaneously the ratio of the backward rate to the forward rate in
the two-message sum-rate can be arbitrarily small.