  In this paper, we study modules for the Twisted Full Toroidal Lie Algebras
(TFTLA). We define a category of bounded modules for TFTLA and classify all the
irreducible modules in that category. A class of irreducible bounded modules
are explicitly constructed by Michael Lau and Yuly Billig using Vertex operator
algebras. We show that they exhaust all the irreducible bounded modules. Our
plan is to describe the top spaces of the bounded irreducible modules for TFTLA
and prove they in fact coincide with the top spaces of the modules constructed
by Lau and Billig. When the top spaces become isomorphic then the corresponding
irreducible modules are also isomorphic.
