One of the key problem in mirror symmetry is that singular fibres in higher dimensional SYZ fibrations of sufficiently interesting spaces (e.g. the quintic 3-fold) have bad singularities. This makes it unreasonable to directly define their Lagrangian Floer cohomology groups, but they are nonetheless expected to support objects of the Fukaya category with self-Floer cohomology isomorphic to the cohomology of a torus. I will describe a method for bypassing this problem by constructing immersed Lagrangians near SYZ singular fibres in three dimensions, with the property that the moduli space of simple objects recovers the mirror space. The construction generalises to higher dimensions.