  We present and analyze $F_\sigma$-Mathias forcing, which is similar but tamer
than Mathias forcing. In particular, we show that this forcing preserves
certain weak subsystems of second-order arithmetic such as $\mathsf{ACA}_0$ and
$\mathsf{WKL}_0 + \mathsf{I}\Sigma^0_2$, whereas Mathias forcing does not. We
also show that the needed reals for $F_\sigma$-Mathias forcing (in the sense of
Blass) are just the computable reals, as opposed to the hyperarithmetic reals
for Mathias forcing.
