  We propose and explore the physics of a toy multiferroic model by coupling
two distinct dipolar XXZ models in transverse fields. We determine first the
rich ground-state phase diagram of the model using density matrix
renormalization group techniques. Then, we explore the dynamics of the system
after global and local quenches, using the time-evolving block decimation
algorithm. After a global quench, the system displays decaying coupled
oscillations of the electric and magnetic spins, in agreement with the
Eigenstate Thermalization Hypothesis (ETH) for many-body interacting quantum
systems. Notably, the spin-spin interactions lead to a sizeable quadratic shift
in the oscillation frequency as the inter-chain coupling is increased. Local
quenches lead to a light-cone-like propagation of excitations. In this case,
the inter-chain coupling drives a transfer of energy between the chains that
generates a novel fast spin-wave mode along the 'magnetic' chain at the speed
of the 'electric' spin-wave. This suggests a limited control mechanism for
faster information transfer in magnetic spin chains using electric fields that
harnesses the electric dipoles as intermediaries.
