There is a set of ideas called mirror symmetry which, roughly speaking, relates symplectic and complex geometry of Calabi--Yau manifolds. There are also extensions to Fano and general type varieties but let's forget about them for now.
There are several more or less precise incarnations of mirror symmetry, including so-called SYZ and HMS programs. Each gives us a certain recipe to construct the manifold mirror to a given Calabi--Yau.
In SYZ, the expectation is that to construct the mirror manifold we need to put our Calabi--Yau in a family of Calabi--Yau's over the punctured disc so that the monodromy operator is maximally unipotent. Different maximally unipotent degenerations can lead to different mirror manifolds.
In HMS, let's assume that we have finally found the right definition of Fukaya category. After certain algebraic manipulations, we can cook up a triangulated category which should be equivalent to the derived category of coherent sheaves on the mirror manifold. If we pick a t-structure in that category, we can Gabriel-style reconstruct the mirror manifold itself from the heart of t-structure. However, if I understand correctly, there is no natural choice of t-structure so we may get several mirror manifolds.
The question is: are these two ambiguities (choosing a maximally unipotent degeneration and choosing a heart of t-structure) in some sort of bijective correspondence? Are there some references where this is studied?
P.S.: to avoid certain subtleties in hyperkaehler geometry, let's assume that the holonomy group of Calabi--Yau's is equal the special unitary group.
EDIT: on second thought, it feels like this should be wrong (Calabi--Yau's can be derived equivalent for reasons that have nothing to do with mirror symmetry). Is there some way to single out the mirrors arising from SYZ inside the set of mirrors arising from HMS?