Let $\cal A$ be an abelian category, say linear over a field, with enough injectives and $\cal P$ be the heart of a t-structure on the bounded derived category $D^b(\cal A )$. Assume that $\cal P$ also has enough injectives. Suppose that the realization functor

$$real:D^b(\cal P)\rightarrow D^b(\cal A)$$

is an equivalence. Now given two objects corresponding through this equivalence, are their Ext-algebras $A_\infty$-quasi-isomorphic?

Edit: If I understand it correctly the realization functor is constructed in BBD as follows:

They take the homotopy category of injective complexes of objects in $\cal A$, equipped with descending filtration, whose filtration steps lie in $\cal P$. This category is called $DF_{bete}$.

Using the boundary map of the triangles $gr^{i+1}F\rightarrow F^{i-1}/F^{i+1} \rightarrow gr^{i} F$ they construct a functor to the category of complexes $DF_{bete}\rightarrow C^b(\cal P)$ and show that it is an equivalence.

Forgetting of the filtration on $DF_{bete}$ translates to a functor $C^b(\cal P)\rightarrow D^b(\cal A)$. The derived functor of this is the realization functor.

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