We show from first principles that the altermagnet α-MnTe reconstructs its magnetic order at the Te-terminated (0001) surface. The ground state has a ferromagnetic outermost Mn bilayer in place of the bulk-continued stacking. The driver is the holes that the surface leaves on the Te mediating the exchange within that bilayer. The computed constant-energy contours agree with photoemission maps of the surface metal. Filling the holes restores the bulk order: the carrier density can control the strength of the surface coupling and, by reversing its sign, the surface magnetic order itself.