For a smooth projective complex variety $X$ of dimension $n$, let $H^i(X)$ denote its integral Hodge structure of weight $i$. Define $\tilde{ H^0}(X) = \bigoplus H^{2i}(X)\otimes \Bbb Z(i)$ and $\tilde{ H^1} (X)= \bigoplus H^{2i+1}\otimes \Bbb Z(i)$, respectively. It is known that any derived equivalence $\Phi:D^b(X)\to D^b(Y)$ induces isomorphisms of rational Hodge structures $\tilde{H^0_\Bbb Q}(X)\cong \tilde{H^0_\Bbb Q}(Y)$ and $\tilde{H^1_\Bbb Q}(X)\cong \tilde{H^1_\Bbb Q}(Y)$, however these isomorphisms are defined by characteristic classes whose coefficients aren't necessarily integral.

Are there any known examples where this isomorphism does not descend to the integral Hodge structures and, furthermore, $\tilde H^*(X)$ and $\tilde H^*(Y)$ are nonisomorphic?