  We propose an adaptive lossy joint source-channel coding (JSCC) scheme for
sending correlated sources over two-terminal discrete-memoryless two-way
channels (DM-TWCs). The main idea is to couple the independent operations of
the terminals via an adaptive coding mechanism, which can mitigate
cross-interference resulting from simultaneous channel transmissions and
concurrently exploit the sources' correlation to reduce the end-to-end
reconstruction distortions. Our adaptive JSCC scheme not only subsumes existing
lossy coding methods for two-way simultaneous communication but also improves
their performance. Furthermore, we derive outer bounds for our two-way lossy
transmission problem and establish complete JSCC theorems in some special
settings. In these special cases, a non-adaptive separate source-channel coding
(SSCC) scheme achieves the optimal performance, thus simplifying the design of
the source-channel communication system.
