  We explore the $C_2$-equivariant spectra $Tmf_1(3)$ and $TMF_1(3)$. In
particular, we compute their $C_2$-equivariant Picard groups and the
$C_2$-equivariant Anderson dual of $Tmf_1(3)$. This implies corresponding
results for the fixed point spectra $TMF_0(3)$ and $Tmf_0(3)$. Furthermore, we
prove a Real Landweber exact functor theorem.
