  Let $\mathsf{X}$ be the product of a complex projective space and a polydisc.
We study Poisson brackets on $\mathsf{X}$ that are log symplectic, that is,
generically symplectic and such that the inverse two-form has only first order
poles. We propose a method of constructing such Poisson brackets that
additionally are elliptic, in a precise sense. Our method relies on the local
Torelli theorem for log symplectic manifolds of Pym, Schedler and the author,
and uses combinatorics of smoothing diagrams. We demonstrate effectiveness of
the method on a series of examples, recovering, in particular, all log
symplectic cases of elliptic Feigin-Odesskii Poisson brackets $q_{n,k}$ on
$\mathbb{P}^{n-1}$.
