  Aims. The main goal of the present paper is to provide the first systematic
numerical study of the propagation of astrophysical relativistic jets, in the
context of high-resolution shock-capturing resistive relativistic
magnetohydrodynamics (RRMHD) simulations. We aim at investigating different
values and models for the plasma resistivity coefficient, and at assessing
their impact on the level of turbulence, the formation of current sheets and
reconnection plasmoids, the electromagnetic energy content, and the dissipated
power. Methods. We use the PLUTO code for simulations and we assume an
axisymmetric setup for jets, endowed with both poloidal and toroidal magnetic
fields, and propagating in a uniform magnetized medium. The gas is assumed to
be characterized by a realistic Synge-like equation of state (Taub equation),
appropriate for such type of astrophysical jets. The Taub equation is combined
here for the first time with the Implicit-Explicit Runge-Kutta time-stepping
procedure, as required in RRMHD simulations. Results. The main result is that
turbulence is clearly suppressed for the highest values of resistivity (low
Lundquist numbers), current sheets are broader, and plasmoids are barely
present, while for low values of resistivity results are very similar to ideal
runs, where dissipation is purely numerical. We find that recipes employing a
variable resistivity based on the advection of a jet tracer or on the
assumption of a uniform Lundquist number improve on the use of a constant
coefficient and are probably more realistic, preserving the development of
turbulence and of sharp current sheets, possible sites for the acceleration of
the non-thermal particles producing the observed high-energy emission.
