Minimum models of second-order set theories

In this article I investigate the phenomenon of minimum models of second-order set theories, focusing on Kelley--Morse set theory $\mathsf{KM}$, G\"odel--Bernays set theory $\mathsf{GB}$, and $\mathsf{GB}$ augmented with the principle of Elementary Transfinite Recursion. The main results are the following. (1) A countable model of $\mathsf{ZFC}$ has a minimum $\mathsf{GBC}$-realization if and only if it admits a parametrically definable global well-order. (2) Countable models of $\mathsf{GBC}$ admit minimal extensions with the same sets. (3) There is no minimum transitive model of $\mathsf{KM}$. (4) There is a minimum $\beta$-model of $\mathsf{GB} + \mathsf{ETR}$. The main question left unanswered by this article is whether there is a minimum transitive model of $\mathsf{GB} + \mathsf{ETR}$.