Let $X$ be a smooth projective variety over a field, than tilting object $T$ on $X$ is a perfect complex that is a compact generator of the derived category $\operatorname{D}(QCoh(X))$ and satisfies condition $Ext^{i}(T, T)=0$ for $i \neq 0$. Tilting algebra $A=End(T)$ is a finite dimensional algebra of finite global dimension. Main result of such geometric tilting theory is an equivalence of triangulated categories $D(X)$ and $D(A)$.

Are there examples of two not isomorphic smooth projective varieties with tilting objects $(X, T)$ and $(X', T')$ such that $End_X(T) \cong End_{X'}(T')$?

In particular this would imply that $\operatorname{D^b}(X) \simeq \operatorname{D^b}(X')$, so by results of Bondal and Orlov $\omega_X$ can't be (anti-)ample, because in this case $X \cong X'$. Moreover, I suppose that if $\omega_X$ is ample than $X$ could not admit a tilting object, but I don't know how to prove this.