  Let $E\mathbb{R}$ be an even-periodic Real Landweber exact $C_2$-spectrum,
and $ER$ its spectrum of fixed points. We compute the $ER$-cohomology of the
infinite stunted projective spectra $P_j$. These cohomology groups combine to
form the $RO(C_2)$-graded coefficient ring of the $C_2$-spectrum $b(ER) =
F(EC_{2+},i_\ast ER)$, which we show is related to $E\mathbb{R}$ by a cofiber
sequence $\Sigma^\sigma b(ER)\rightarrow b(ER)\rightarrow E\mathbb{R}$. We
illustrate our description of $\pi_\star b(ER)$ with the computation of some
$ER$-based Mahowald invariants.
