We consider the free propagation of totally symmetric massive bosonic fields in nontrivial backgrounds. The mutual compatibility of the dynamical equations and constraints in flat space amounts to the existence of an Abelian algebra formed by the dAlembertian, divergence and trace operators. The latter, along with the symmetrized gradient, symmetrized metric and spin operators, actually generate a bigger non-Abelian algebra, which we refer to as the consistency algebra. We argue that in nontrivial backgrounds, it is some deformed version of this algebra that governs the consistency of the system. This can be motivated, for example, from the theory of charged open strings in a background gauge field, where the Virasoro algebra ensures consistent propagation. For a gravitational background, we outline a systematic procedure of deforming the generators of the consistency algebra in order that their commutators close. We find that equal-radii AdSp X Sq manifolds, for arbitrary p and q, admit consistent propagation of massive and massless fields, with deformations that include no higher-derivative terms but are non-analytic in the curvature. We argue that analyticity of the deformations for a generic manifold may call for the inclusion of mixed-symmetry tensor fields like in String Theory.
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