Dissipative processes can fundamentally alter magnetic order in quantum spin chains. In this work, we introduce a non-perturbative analytic mapping framework that enables the systematic engineering of magnetic phases by controlling the locality of the attached bath. 

Our approach reveals how spin-bath interactions suppress spin splittings, induce bath dressing, and lead to the mixing of spin-spin couplings. Notably, it uncovers the emergence of effective non-local ferromagnetic interactions between spins coupled to a common bath, interactions that become fully long-ranged when the bath is global. 

This general mapping technique is broadly applicable across spin models. We demonstrate: (i) a bath-induced transition from antiferromagnetic (AFM) to ferromagnetic order in a Heisenberg spin chain, (ii) a crossover from AFM to extended Néel-phase order in a transverse-field Ising chain with pairwise bath couplings, and (iii) a quantum phase transition in the fully connected Ising model. 

Finally, we demonstrate how the mapping approach can be applied to higher dimensions, larger spin systems, and fermionic systems.