I am interested in $P$ that is smooth and proper over a field and such that the derived category of coherent sheaves $D^b(P)$ possesses a $t$-structure whose heart is the category of $R'$-modules for some (non-commutative) ring $R'$.

Which examples are known? Do all of them support full exceptional collections (see https://en.wikipedia.org/wiki/Semiorthogonal_decomposition#Exceptional_collection)?

This question is close to Bounded derived categories of which smooth projectives possess bounded t-structures whose hearts have enough injectives?