We study a stable partial matching $τ$ of the (possibly randomized) $d$-dimensional lattice with a stationary determinantal point process $Ψ$ on $\mathbb{R}^d$ with intensity $α>1$. For instance, $Ψ$ might be a Poisson process. The matched points from $Ψ$ form a stationary and ergodic (under lattice shifts) point process $Ψ^τ$ with intensity $1$ that very much resembles $Ψ$ for $α$ close to $1$. On the other hand $Ψ^τ$ is hyperuniform and number rigid, quite in contrast to a Poisson process. We deduce these properties by proving more general results for a stationary point process $Ψ$, whose so-called matching flower (a stopping set determining the matching partner of a lattice point) has a certain subexponential tail behaviour. For hyperuniformity, we also additionally need to assume some mixing condition on $Ψ$. Further, if $Ψ$ is a Poisson process then $Ψ^τ$ has an exponentially decreasing truncated pair correlation function.