They propose the reverse approach. Take one of the simplest low-energy phenomenological models compatible with your observations - such as the MSSM. Define its gauge-invariant monomials to parameterize a moduli space; these are used as F-terms or D-terms. Determine the dimensionality and topology of the resulting moduli space. And find a string model that exactly matches it.
They often mention that they would like to derive that the moduli space is a Calabi-Yau three-fold itself. I find it a bit exaggerated. The Calabi-Yau space can only be a moduli space at low energies if you consider something like one D3-brane on Calabi-Yaus in type IIB - but there are really no phenomenologically viable models of this kind. Note that the 3-fold in the F-theory flux constructions is not a Calabi-Yau manifold.
In reality, the moduli spaces at low energies describe the moduli spaces of shapes of manifolds such as the Calabi-Yaus or moduli spaces of gauge bundles over them. They can have many different dimensionalities and topologies. In the top-down approach, we know very well what is a natural requirement for a model to be phenomenologically appealing: we want to get as close to the Standard Model or MSSM as possible, and remove all exotics.
A corresponding task in their bottom-up approach is not quite determined, as far as I can see. What properties should the low-energy moduli spaces derived by their algorithm have in order to tell us that it looks like it comes from string theory? I think that the idea that it should be a Calabi-Yau manifold is naive, and I don't have any better replacement for this proposed answer.
For example, they seem very excited by having obtained a moduli space whose Hodge numbers coincide with those of a CP^2:
and otherwise zero. I personally have not understood why this "simple" Hodge diamond is more attractive than other Hodge diamonds that they could have derived.