  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.
