The Gale correspondence provides a duality between sets of n general points in projective spaces $P^s$ and $P^r$ when $n$ equals $r + s + 2$. By a result of Mukai, the blow-up of $P^4$ at $8$ points say $X$, can be realized as a moduli space of torsion-free rank $2$ semi-stable sheaves (with certain fixed Chern class datum) on the blow-up of $P^2$ in $8$ Gale dual points. In a recent work, Casagrande, Codogni and Fanelli use this to describe the Mori chamber decomposition of the effective cone of divisors of $X$. It was shown by Castravet and Tevelev that the blow-up of $P^r$ at $n$ points for the case when $r\geq 5$ and $n\geq r + 4$ is no longer a Mori dream space. In joint work with Carolina Araujo, Ana-Maria Castravet and Diletta Martinelli we show that even in this case it is possible to give a Mori chamber type decomposition for a part of the effective cone.