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If something causes your unit to become battle shocked during your opponents turn like Tyranid "shadow of the warp" or Infernus marines "purge the foe", come your next command phase if your squad is still above half strength does it have to take a battle shock test to "rally" or does it recover automatically?

No. Battle-shock for your units last until the beginning of your next Command Phase. Come the Battle-shock step of the Command Phase, none of your units are battle-shocked and only those that are below half starting strength have to test (barring other rules interactions).