  Let $J$ be any ideal in a strongly $F$-regular, diagonally $F$-split ring $R$
essentially of finite type over an $F$-finite field. We show that $J^{s+t}
\subseteq \tau(J^{s - \epsilon}) \tau(J^{t-\epsilon})$ for all $s, t, \epsilon
> 0$ for which the formula makes sense. We use this to show a number of novel
containments between symbolic and ordinary powers of prime ideals in this
setting, which includes all determinantal rings and a large class of toric
rings in positive characteristic. In particular, we show that $P^{(2hn)}
\subseteq P^n$ for all prime ideals $P$ of height $h$ in such rings.
