In many systems, including sedimenting particles and bird flocks, interactions do not derive from a potential and are generally non-reciprocal, meaning that they do not obey the action–reaction principle. As a result, one cannot define a conventional energy function or use analytical and numerical tools that rely on it. Here we address this limitation by constructing a Hamiltonian with auxiliary degrees of freedom that, under a constraint, generates the original non-reciprocal dynamics. We show that Monte Carlo simulations based on the constrained Hamiltonian reproduce both stationary and non-stationary states of the original Langevin dynamics, as we illustrate for dissipative XY spins with vision-cone interactions. The symplectic structure inherent to the construction also lets us apply established ideas from Hamiltonian engineering, which we demonstrate by varying the amplitude of a periodic (Floquet) drive to tune the spin interactions between square- and chain-lattice geometries. Overall, our construction paves the way towards extending statistical mechanics and Hamiltonian dynamics to non-reciprocal systems.