Our aim is to formulate hydrodynamicslike theory for the fluids for which the classical hydrodynamics fails (e.g., polymeric fluids). In addition, we limit ourselves in this paper to the fluids for which the enlarged set of classical hydrodynamic fields, enlarged by the fields of the extra stress tensor and the extra energy flux, represent a dynamically closed set of state variables. We say, roughly speaking, that a set of state variables is dynamically closed if predictions calculated from the dynamical theory that uses this set of state variables agree, to some extent, with results of hydrodynamicslike (rheological) observations. Examples of such fluids can be found in Jou et al., [Extended Irreversible Thermodynamics (Springer, Berlin, 1996)]. In this book the hydrodynamicslike theory whose consequences are compared with results of observations is linear in the fields that extend the set of classical hydrodynamic fields. In this paper we extend the linear theory to a fully nonlinear theory. The additional physical insight that makes the extension possible is the requirement of a generalized Hamiltonian structure. This structure has been identified in all dynamical theories (on all levels of description, including, for example, kinetic theory) that describe the time evolution of externally unforced fluids (i.e., fluids that eventually reach equilibrium states at which they can be well described by equilibrium thermodynamics). A prominent new feature of the nonlinear theory is that the extra fields extending the set of classical hydrodynamical fields are not exactly the fields of the extra stress and the extra energy flux, but new fields from which the extra stress and the extra energy flux can always be calculated. The inverse of this map exists, however, always only in the linear case.