  Radiation-reaction forces originating from the emission of gravitational
waves (GW) bring binaries to close proximity and are thus responsible for
virtually all the mergers that we can observe in GW interferometers. We show
that there exists a supplementary radiation-reaction force between two binaries
interacting gravitationally, changing in particular the decay rate of the
semimajor axis under the emission of GW. This new binary-binary force is in
some settings of the same order-of-magnitude than the usual 2.5PN force for an
isolated binary and presents some striking features such as a dependence on
retarded time even in the post-Newtonian regime where all velocities are
arbitrarily small. Using Effective Field Theory tools, we provide the
expression of the force in generic configurations and show that it interpolates
between several intuitive results in different limits. In particular, our
formula generalizes the standard post-Newtonian estimates for
radiation-reaction forces in $N$-body systems which are valid only in the limit
where the GW wavelength goes to infinity.
