We give a construction of Lagrangian torus fibrations with controlled discriminant locus on certain affine varieties. In particular, we apply our construction in the following ways: > We find a Lagrangian torus fibration on the 3-fold negative vertex whose discriminant locus has codimension 2; this provides a local model for finding torus fibrations on compact Calabi–Yau 3-folds with codimension 2 discriminant locus. > We find a Lagrangian torus fibration on a neighbourhood of the one-dimensional stratum of a simple normal crossing divisor (satisfying certain conditions) such that the base of the fibration is an open subset of the cone over the dual complex of the divisor. This can be used to construct an analogue of the non-archimedean SYZ fibration constructed by Nicaise, Xu and Yu.