We study a $2d$ Hamiltonian fluid made of particles carrying spins coupled to their velocities. At low temperatures and intermediate densities, this conservative system exhibits phase coexistence between a collectively moving droplet and a still gas. The particle displacements within the droplet have remarkably similar correlations to those of birds flocks. The center of mass behaves as an effective self-propelled particle, driven by the droplets total magnetization. The conservation of a generalized angular momentum leads to rigid rotations, opposite to the fluctuations of the magnetization orientation that, however small, are responsible for the shape and scaling of the correlations.
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