  This paper develops a general methodology for a posteriori error estimation
in time-dependent multiphysics numerical simulations. The methodology builds
upon the generalized-structure additive Runge--Kutta (GARK) approach to time
integration. GARK provides a unified formulation of multimethods that simulate
complex systems by applying different discretization formulas and/or different
time steps to individual components of the system. We derive discrete GARK
adjoints and analyze their time accuracy. Based on the adjoint method, we
establish computable a posteriori identities for the impacts of both temporal
and spatial discretization errors on a given goal function. Numerical examples
with reaction-diffusion systems illustrate the accuracy of the derived error
measures. Local error decompositions are used to illustrate the power of this
framework in adaptive refinements of both temporal and spatial meshes.
