Auxiliary Government is an upgrade for the TEC Argonev Star Base. It is functionally identical to its Advent and Vasari equivalents, Enduring Devotion for the Transcencia Star Base and Enforced Loyalty for the Orkulus Star Base respectively.
When a starbase has this upgrade, you cannot lose control of the planet it protects. Even if the planet is bombarded down to 0 hit points, you will not lose control. This means the enemy must first destroy the starbase before they can take control of the planet. This upgrade will not affect allied or enemy planets, it only affects your own planets. If the planet turns neutral, however, this upgrade will prevent anyone but you from colonizing the planet afterwards, and once colonized this upgrade will protect the new planet until the starbase is destroyed.
This upgrade is extremely useful if you're using a starbase to protect your homeworld, and is one of the only ways to stop Novalith Cannons from destroying your planets. This upgrade is critical in multiplayer, particularly for TEC and Advent, since human players will often just ignore an immobile starbase and move to bombard the planet. Using this upgrade, you can place the starbase anywhere in the gravity well and it will always protect the planet. The enemy is still capable of bombarding the planet to reduce population and tax income, however.
Note that the upgrade will not prevent enemy culture from turning your planet neutral however, but it will prevent enemies from colonizing said neutral planet until the starbase has been destroyed. But because of this, it means that starbases alone cannot defend against concentrated Deliverance Engine fire, with the possible exception of a Transcencia using Evangelization Nodes and/or Induced Reverence.
The Auxiliary Government upgrade has largely made the Shield Generator structure obsolete in Entrenchment, since it appears earlier in the technology tree and a starbase is much more difficult to destroy than a little orbital generator.