For which smooth projective $P$ over a field there exists a bounded $t$-structure $t$ on the bounded derived category of coherent sheaves $D^b(P)$ such the heart $Ht$ of $t$ has enough injectives? Note that the Verdier dual of an example of this sort will give a $t$-structure $t'$ whose heart $Ht'$ will have enough projectives.

I believe that my weight structure arguments give some examples if $P$ possesses a full exceptional collection (see https://en.wikipedia.org/wiki/Semiorthogonal_decomposition#Exceptional_collection). So I wonder whether there exists examples for $P$ that does not support a collection of this sort.

Conversely, what can one say about $P$ knowing that a $t$-structure of this sort exists? Will it help if one assumes that $Ht$ is a "nice category of modules" (in a certain sense; yet the corresponding ring is not commutative in general).