  In this paper, we first construct the twisted full toroidal Lie algebra by an
extension of a centreless Lie torus $LT$ which is a multiloop algebra twisted
by several automorphisms of finite order and equipped with a particular
grading. We then provide a complete classification of all the irreducible
integrable modules with finite dimensional weight spaces for this twisted full
toroidal Lie algebra having a non-trivial $LT$-action and where the centre of
the underlying Lie algebra acts trivially.
