$F$-theory over a Fano threefold built from $A_4$-roots

In a previous paper, the authors showed the advantages of building a $\mathbb{Z}_2$-action into an $F$-theory model $W_4 / B_3$, namely the action of complex conjugation on the complex algebraic group with compact real form $E_8$. The goal of this paper is to construct the Fano threefold $B_3$ directly from the roots of $SU(5)$ in such a way that the action of complex conjugation is exactly the desired $\mathbb{Z}_2$-action and the quotient of this action on $W_4 / B_3$ and its Heterotic dual have the phenomenologically correct invariants.