An infinite family of strongly real Beauville $p$-groups

We give an infinite family of non-abelian strongly real Beauville $p$-groups for every prime $p$ by considering the quotients of triangle groups, and indeed we prove that there are non-abelian strongly real Beauville $p$-groups of order $p^n$ for every $n\ge 3,5$ or $7$ according as $p\ge 5$ or $p=3$ or $p =2$\mathrm{.}ThisshowsthattherearestronglyrealBeauville$p$-groups exactly for the same orders for which there exist Beauville $p$-groups.