  We examine the noncommutative minimal model program for orders on arithmetic
surfaces, or equivalently, arithmetic surfaces enriched by a Brauer class
$\beta$. When $\beta$ has prime index $p>5$, we show the classical theory
extends with analogues of existence of terminal resolutions, Castelnuovo
contraction and Zariski factorisation. We also classify $\beta$-terminal
surfaces and Castelnuovo contractions, and discover new unexpected behaviour.
