Phantom depth and flat base change

We prove that if $f: (R,\mathfrak{m}) \rightarrow (S,\mathfrak{n})$ is a flat local homomorphism, $S/\mathfrak{m} S$ 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.