The augmented Langevin approach described in a previous article is applied to the problem of introducing multiplicative noise and nonlinear dissipation into an arbitrary Hamiltonian system in a thermodynamically consistent way, so that a canonical equilibrium distribution is approached asymptotically at long times. This approach leads to a general nonlinear fluctuation-dissipation relation which, for a given form of the multiplicative noise (chosen on physical grounds), uniquely determines the form of the nonlinear dissipative terms needed to balance the fluctuations. In addition to the noise and dissipation terms, the augmented Langevin equation contains an additional term whose form depends on the stochastic interpretation rule used. This term vanishes when the Stratonovich rule is chosen and the noise itself is of a Hamiltonian origin. This development provides a simple phenomenological route to results previously obtained by detailed analysis of microscopic system-bath models. The procedure is illustrated by applications to a mechanical oscillator with fluctuating frequency, a classical spin in a fluctuating magnetic field, and the Brownian motion of a rigid rotor.