  In this paper, we study the elliptic spectral sequence computing
$tmf_*(\mathbb{R} P^2)$ and $tmf_* (\mathbb{R} P^2 \wedge \mathbb{C} P^2)$.
Specifically, we compute all differentials and resolve exotic extensions by 2,
$\eta$, and $\nu$. For $tmf_* (\mathbb{R} P^2 \wedge \mathbb{C} P^2)$, we also
compute the effect of the $v_1$-self maps of $\mathbb{R} P^2 \wedge \mathbb{C}
P^2$ on $tmf$-homology.

Algebraic Topology