  In this work we study the moduli space of instanton bundles on the flag
twistor space $F:=F(0,1,2)$. We stratify them in terms of the minimal twist
supporting global sections and we introduce the notion of (special) 't Hooft
bundle on $F$. In particular we prove that there exist $\mu$-stable 't Hooft
bundles for each admissible charge $k$. We completely describe the geometric
structure of the moduli space of (special) 't Hooft bundles for arbitrary
charge $k$. Along the way to reach these goals, we describe the possible
structures of multiple curves supported on some rational curves in $F$ as well
as the moduli space of del Pezzo surfaces realized as hyperplane sections of
$F$. Finally we investigate the splitting behaviour of 't Hooft bundles when
restricted to conics.
