Odd fake $\mathbb{Q}$ -homology quadrics exist

We show the existence of odd fake $\mathbb{Q}$-homology quadrics, namely of minimal surfaces $S$ of general type which have the same $\mathbb{Q}$-homology as a smooth quadric $Q\cong ({\mathbb{P}}^1(\mathbb{C}){)}^2$, but have an odd intersection form on $H^2(S,\mathbb{Z})\mathrm{/}Tors(S)$, where $Tors(S)$ is the Torsion subgroup. Our examples are provided by a special 1-dimensional family of surfaces isogenous to a product of unmixed type.