  We present an analytic approach to treat open quantum systems strongly
coupled to multiple environments via noncommuting system operators, a prime
example is a qubit concurrently coupled to both decoherring and dissipative
baths. Our approach, which accommodates strong system-bath couplings,
generalizes the recently developed reaction-coordinate polaron transform method
[PRX Quantum ${\bf 4}$, 020307 (2023)] to handle couplings to baths via
noncommuting system operators. This approach creates an effective Hamiltonian
that reveals the cooperative effect of the baths on the system. For a spin
impurity coupled to both dissipative and decoherring environments, the
effective Hamiltonian predicts the suppression of relaxation by decoherence --
a phenomenon previously observed in simulations but lacking a theoretical
foundation. We also apply the method to an ensemble of spins coupled to local
baths through noncommuting operators, demonstrating the engineering of the
Kitaev XY spin chain interaction. Noncommutativity is a feature of quantum
systems; future prospects of our approach include the study of thermal machines
that leverage such genuine quantum effects.
