Concrete based additive manufacturing in civil engineering poses several challenges to its modelling and simulation. Not only is the material very heterogeneous with a very wide range of often uncertain material properties, the modelling also has to cope with different phases involved (i.e. cement paste, aggregates and air) which differ substantially in their properties. In this talk we will discuss the challenges arising from the complex rheology of the cement past and the large density and viscosity ratios between the cement past and the surrounding air. Due to its kinetic nature, the lattice Boltzmann method is able to model the yield stress behaviour of the cement past without regularization. This is possible since the kinetic approach models viscosity through a collision frequency which is proportional to the inverse of the diffusion coefficient. The numerically infeasible infinite viscosity state of unyielded fluid is hence replaced by a numerical feasible zero collision frequency state. The large density ratio between the cement past and the air can be problematic for the lattice Boltzmann method. In some additive manufacturing techniques, like e. g. material extrusion, the air does not significantly impact the dynamics of the printing process. However, the accidental inclusion of air bubbles will still diminish the quality of the printed parts. In shotcrete 3D printing the deposition of material is driven by an expanding air stream such that the air phase becomes an essential aspect for the modelling of the process. In the lattice Boltzmann method, multiphase flows can be modelled with either diffuse or sharp interface models. Diffusive interface models often have difficulties with the combination of large density ratios and large viscosity ratios between the phases. Sharp interface models usually neglect the lighter phase altogether and replace the interphase by a free surface. Here we discuss how the advantages from both concepts can be combined to model both phases at a large albeit finite density ratio.