For a determinantal point process $X$ with a kernel $K$ whose spectrum is strictly less than one, André Goldman has established a coupling to its reduced Palm process $X^u$ at a point $u$ with $K(u,u)>0$ so that in distribution $X^u$ is obtained by removing a finite number of points from $X$. The intensity function of the difference $X\setminus X^u$ is known, but apart from special cases the distribution of $X\setminus X^u$ is unknown. Considering the restriction $X_S$ of $X$ to any compact set $S$, we establish a coupling of $X_S$ and its reduced Palm process $X^u_S$ so that the difference is at most one point. Specifically, we assume $K$ restricted to $S\times S$ is either (i) a projection or (ii) has spectrum strictly less than one. In case of (i), we have in distribution that $X^u_S$ is obtained by removing one point from $X_S$, and we can specify the distribution of this point. In case of (ii), in distribution we obtain $X^u_S$ either by moving one point in $X$ or by removing one point from $X_S$, and to a certain extent we can describe the distribution of these points. We discuss how Goldman's and our results can be used for quantifying repulsiveness in $X$.

Keywords: Ginibre point process, globally most repulsive determinantal point process, isotropic determinantal point process on the sphere, globally most repulsive determinantal point process, projection kernel, stationary determinantal point process in Euclidean space.