  Tidal dissipation plays an important role in the dynamical evolution of
moons, planets, stars and compact remnants. The interesting complexity
originates from the interplay between the internal structure and external tidal
forcing. Recent and upcoming observing missions of exoplanets and stars in the
Galaxy help to provide constraints on the physics of tidal dissipation. It is
timely to develop new N-body codes, which allow for experimentation with
various tidal models and numerical implementations. We present the open-source
N-body code TIDYMESS, which stands for ``TIdal DYnamics of Multi-body
ExtraSolar Systems''. This code implements a creep deformation law for the
bodies, parametrized by their fluid Love numbers and fluid relaxation times.
Due to tidal and centrifugal deformations, we approximate the general shape of
a body to be an ellipsoid. We calculate the associated gravitational field to
quadruple order, from which we derive the gravitational accelerations and
torques. The equations of motion for the orbits, spins and deformations are
integrated directly using a fourth-order integration method based on a
symplectic composition. We implement a novel integration method for the
deformations, which allows for a time step solely dependent on the orbits, and
not on the spin periods or fluid relaxation times. This feature greatly speeds
up the calculations, while also improving the consistency when comparing
different tidal regimes. We demonstrate the capabilities and performance of
TIDYMESS, particularly in the niche regime of parameter space where orbits are
chaotic and tides become non-linear.
