  We introduce a novel reservoir engineering approach for stabilizing
multi-component Schr\"odinger's cat manifolds. The fundamental principle of the
method lies in the destructive interference at crossings of gain and loss
Hamiltonian terms in the coupling of an oscillator to a zero-temperature
auxiliary system, which are nonlinear with respect to the oscillator's energy.
The nature of these gain and loss terms is found to determine the rotational
symmetry, energy distributions, and degeneracy of the resulting stabilized
manifolds. Considering these systems as bosonic error-correction codes, we
analyze their properties with respect to a variety of errors, including both
autonomous and passive error correction, where we find that our formalism gives
straightforward insights into the nature of the correction. We give example
implementations using the anharmonic laser-ion coupling of a trapped ion
outside the Lamb-Dicke regime as well as nonlinear superconducting circuits.
Beyond the dissipative stabilization of standard cat manifolds and novel
rotation symmetric codes, we demonstrate that our formalism allows for the
stabilization of bosonic codes linked to cat states through unitary
transformations, such as quadrature-squeezed cats. Our work establishes a
design approach for creating and utilizing codes using nonlinearity, providing
access to novel quantum states and processes across a range of physical
systems.
