Let $\mathcal A$ and $\mathcal B$ be abelian categories (with enough injectives and countable products) and $$F:D^+(\cal A) \rightarrow D^+(\cal B)$$ be a triangulated functor. I am interested whether there exists a dg-lift, by which I mean dg-functor $$\tilde F:C^+(inj \cal A)\rightarrow C^+( inj \cal B)$$ between the dg-categories of bounded below complexes with injective entries and usual $Hom^\bullet$-complexes as morphisms, which induces $F$ on homotopy categories. I am also interested in variants with $D^+$ replaced by some other nice subcategory of the derived category, projectives instead of injectives etc.

Now I have heard from time to time the statement, that

"Any functor encountered in practice has a dg- lift. "

Yet I am not really aware of criteria which guarantee dg-lifts, I only know one recipe to construct dg-lifts:

First step: Try to lift F to a dg-functor $$C^+(inj \cal A)\rightarrow C^+(\cal B)$$ This step is often but not always obvious. For example it works when $F$ is a derived functor. And it is not obvious to me for example if $\cal A$ is the heart of a t-structure on $D^b(\cal B)$ and $F$ is the realization functor: $real:D^b(\cal A)\rightarrow D^b(\cal B)$.

Second step: Hope that injective resolutions can be choosen in a nice way, giving a dg-functor $$C^+(\cal B)\rightarrow C^+(inj \cal B)$$ and compose this functor with the one from step 1. This should for example work if $\cal B$ is the category of modules over a sheaf of k-algebras (where k is a field) and maybe doesn't if $\cal B$ is the category of abelian groups.

Are there other methods/criteria that allow to construct dg-lifts? I m especially interested in situations where step 1 is not obvious.

What are examples of such functors $F$ which do **not** have dg-lifts, even if $\cal A$ and $\cal B$ have enough injectives/projectives? How does one proof such a statement?