Phantom depth and flat base change

We prove that if \(f: (R,\m) \to (S,\n)\) is a flat local homomorphism, \(S/\m S\) is Cohen-Macaulay and \(F\)-injective, and \(R\) and \(S\) share a weak test element, then a tight closure analogue of the (standard) formula for depth and regular sequences across flat base change holds. As a corollary, it follows that phantom depth commutes with completion for excellent local rings. We give examples to show that the analogue does not hold for surjective base change.