Let $\mathcal{C}$ be a nice $k$-linear abelian category (the example I have in mind is the category of coherent sheaves on a smooth projective variety over $\mathbb{C}$). Let $B \in D^{b}(\mathcal{C})$ such that the DG-algebra $\mathrm{RHom}(B,B)$ is formal (that is quasi-isomorphic, as a DG-algebra to its cohomology algebra).

I would like to know if there exists a complex $B^{\bullet}$ representing $B$ such that: $$\mathrm{Hom}^{\bullet}(B^{\bullet}, B^{\bullet}) = \mathrm{Ext}^{\bullet}(B,B),$$ where the sign equal really means equality (and not just quasi-isomorphism of DG algebras)?

Here $\mathrm{Hom}^{\bullet}(B^{\bullet}, B^{\bullet})$ is the DG-algebra defined by: _$\mathrm{Hom}^{k}(B^{\bullet}, B^{\bullet})$ is the vector space of sequences of maps $ f^p : B^{p} \rightarrow B^{p+k}$,

_the multiplication map in $\mathrm{Hom}^{\bullet}(B^{\bullet}, B^{\bullet})$ is the composition of maps,

_the differential is the one inherited from that of $B^{\bullet}$.

Thanks a lot!