  The competition between interactions and dissipative processes in a quantum
many-body system can drive phase transitions of different order. Exploiting a
combination of cluster methods and quantum trajectories, we show how the
systematic inclusion of (classical and quantum) nonlocal correlations at
increasing distances is crucial to determine the structure of the phase
diagram, as well as the nature of the transitions in strongly interacting spin
systems. In practice, we focus on the paradigmatic dissipative quantum Ising
model: in contrast to the non-dissipative case, its phase diagram is still a
matter of debate in the literature. When dissipation acts along the interaction
direction, we predict important quantitative modifications of the position of
the first-order transition boundary. In the case of incoherent relaxation in
the field direction, our approach confirms the presence of a second-order
transition, while does not support the possible existence of multicritical
points. Potentially, these results can be tested in up-to date quantum
simulators of Rydberg atoms.
