  We extend the Post-Minkowskian (PM) effective field theory (EFT) approach to
incorporate conservative and dissipative radiation-reaction effects in a
unified framework. This is achieved by implementing the Schwinger-Keldysh
"in-in" formalism and separating conservative and non-conservative terms
according to the formulation in [1210.2745], which we show promotes Feynman's
$i0$-prescription and cutting rules to a prominent role at the classical level.
The resulting integrals, involving both Feynman and retarded propagators, can
be bootstrapped to all orders in the velocity via differential equations with
boundary conditions including potential and radiation modes. As a paradigmatic
example we provide an ab initio derivation of the classical solution to the
scattering problem in general relativity to ${\cal O}(G^3)$. For the sake of
completeness, we also reproduce the leading order radiation-reaction effects in
classical electrodynamics.
