We study the Rayleigh-Taylor instability (RTI) at a prominence-corona transition region in a non-linear regime. Our aim is to understand how the presence of neutral atoms in the prominence plasma influences the instability growth rate, as well as the evolution of velocity, magnetic field vector, and thermodynamic parameters of turbulent drops. We perform 2.5D numerical simulations of the instability initiated by a multi-mode perturbation at the corona-prominence interface using a single-fluid magnetohydrodynamic (MHD) approach including a generalized Ohm's law. The initial equilibrium configuration is purely hydrostatic and contains a homogeneous horizontal magnetic field forming an angle with the direction in which the plasma is perturbed. We analyze simulations with two different orientations of the magnetic field. For each field orientation we compare two simulations, one for the pure MHD case, and one including the ambipolar diffusion in Ohm's law (AD case). Other than that, both simulations for each field orientation are identical. The numerical results in the initial stage of the instability are compared with the analytical linear calculations. We find that the configuration is always unstable in the AD case. The growth rate of the small-scale modes in the non-linear regime is up to 50% larger in the AD case than in the purely MHD case and the average velocities of flows are a few percentage points higher. Significant drift momenta are found at the interface between the coronal and the prominence material at all stages of the instability, produced by the faster downward motion of the neutral component with respect to the ionized component. The differences in temperature of the bubbles between the ideal and non-ideal case are also significant, reaching 30%. There is an asymmetry between large rising bubbles and small-scale down flowing fingers, favoring the detection of upward velocities in observations.