  We build a set of new observables using closely related non-leptonic
penguin-mediated $B_d$ and $B_s$ decays: ${\bar B}_{d,s}\to
K^{*0}\bar{K}^{*0}$, ${\bar B}_{d,s}\to K^{0}\bar{K}^{0}$, ${\bar B}_{d,s}\to
K^{0}\bar{K}^{*0}$ and ${\bar B}_{d,s}\to\bar{K}^{0}{K^{*0}}$ together with
their CP conjugate partners. These optimised observables are designed to reduce
hadronic uncertainties, mainly coming from form factors and power-suppressed
infrared divergences, and thus maximize their sensitivity to New Physics (NP).
The deviations observed with respect to the SM in the ratios of branching
ratios of ${\bar B}_{d,s}\to K^{*0}\bar{K}^{*0}$ ($2.6\sigma$) and ${\bar
B}_{d,s}\to K^{0}\bar{K}^{0}$ ($2.4\sigma$) can be explained by simple NP
scenarios involving the Wilson coefficients ${\cal C}_4$ and ${\cal C}_6$ (QCD
penguin operators) and the coefficient ${\cal C}_{8g}$ (chromomagnetic
operator). The optimised observables for ${\bar B}_{d,s}\to K^{0}\bar{K}^{*0}$
and ${\bar B}_{d,s}\to\bar{K}^{0}{K^{*0}}$ show distinctive patterns of
deviations with respect to their SM predictions under these NP scenarios. The
pattern of deviations of individual branching ratios, though affected by
significant hadronic uncertainties, suggests that NP is needed both in $b\to d$
and $b\to s$ transitions. We provide the regions for the Wilson coefficients
consistent with both optimised observables and individual branching ratios. The
NP scenarios considered to explain the deviations of ${\bar B}_{d,s}\to
K^{*0}\bar{K}^{*0}$ and ${\bar B}_{d,s}\to K^{0}\bar{K}^{0}$ can yield
deviations up to an order of magnitude among the observables that we introduced
for ${\bar B}_{d,s} \to K^{0} \bar{K}^{*0}$ and ${\bar B}_{d,s}\to\bar{K}^{0}
{K^{*0}}$. Probing these new observables experimentally may confirm the
consistency of the deviations already observed and provide a highly valuable
hint of NP in the non-leptonic sector.
