Let's take attrition for example. If I have CUs in an independent space during the attrition phase, will my CUs suffer attrition?
Or, when deciding on political consequences of winning a battle, take this example: You are the Samnite/Etruscan player with Samnite CUs in an Etruscan home space. You use another Samnite army to defeat a Gallic army. Can you use a point of support to remove the independent PC marker from your Etruscan ally's home space?