For multi-phase flow in porous media, the fluids will experience reduced conductivity due to fluid-fluid interactions. This effect is modeled by the so-called relative permeability. The relative permeability is commonly assumed to be a scalar quantity, even though justifications for this modeling choice seldom are presented. In this paper, we show that the relative permeability can yield preferential flow directions, and that furthermore, these directions may vary as a function of the fluids present. To model these effects properly, the relative permeability should be considered a tensor. As shown in the paper, standard numerical methods cannot simulate tensor relative permeabilities. We therefore propose two new methods to remedy the situation. One scheme can be seen as an attempt to amend traditional methods to handle tensor relative permeabilities. The second scheme is a consistent approach to simulate the tensors. We also present numerical simulations of cases where upscaling leads to tensor relative permeabilities. The results show that the new methods are indeed capable of capturing tensor effects, moreover, they are in very good agreement with fine scale reference solutions.