We consider the nonequilibrium dynamics of a driven dissipative spin chain with chiral coupling to a one-dimensional (1D) bosonic bath, and its atomic implementation with a two-species mixture of cold quantum gases. The reservoir is represented by a spin-orbit coupled 1D quasicondensate of atoms in a magnetized phase, while the spins are identified with motional states of a separate species of atoms in an optical lattice. The chirality of reservoir excitations allows the spins to couple differently to left-and right-moving modes, which in our atomic setup can be tuned from bidirectional to purely unidirectional. Remarkably, this leads to a pure steady state in which pairs of neighboring spins form dimers that decouple from the remainder of the chain. Our results also apply to current experiments with two-level emitters coupled to photonic waveguides.PACS numbers: 03.65. Yz, 67.85.Jk, 42.50.Dv, 03.67.Bg In an open quantum many-body system, the competition of particle interactions, external driving and the dissipative coupling to a quantum reservoir can result in novel scenarios for the formation of strongly correlated quantum states [1]. This is not only of interest as a nonequilibrium condensed matter problem per se [2-9], but dissipatively prepared entangled states also provide a potential resource for quantum information tasks [10][11][12][13][14][15]. Quantum optical systems of cold atoms or solid-state impurities provide a natural setting for such open manybody quantum systems. The paradigmatic example is given by an ensemble of two-level atoms driven by laser light, and coupled to a photonic reservoir [16][17][18][19], e.g., as one-dimensional (1D) engineered photonic band gap materials [20]. These model systems can be described as a collection of spin-1/2 systems, which via the photonic modes interact with long-range dipole-dipole interactions, and exhibit collective and enhanced decay into radiation modes of photonic structures. The realization of such Dicke-type models [21, 22] coupled to low-dimensional quantum reservoirs, and the observation of the associated dynamical quantum phases and phase transitions are, at present, an outstanding challenge in quantum optics [23][24][25][26].In the present work, we introduce a realization of dissipative quantum magnetism based on cold atoms in optical lattices [27,28], where the quantum reservoir is represented by phononic degrees of freedom of a 1D spin-orbit coupled Bose-Einstein quasicondensate (quasi-BEC) [29][30][31][32][33][34][35]. This model system provides a faithful and experimentally realistic representation of a chain of driven spin-1/2 particles coupled to a 1D bosonic bath. Crucially, spin-orbit coupling (SOC) makes the reservoir chiral, with the spins coupling differently to the left and right propagating modes, γ L = γ R [cf. Fig. 1(a)]. This asymmetry is, moreover, tunable via the atomic parameters, making it possible to engineer the spin-bath coupling from purely unidirectional to fully bidirectional.To describe the dynamics of our 1D spin c...