  In this paper, we develop the theory of flag manifold over a semifield for
any Kac-Moody root datum. We show that the flag manifold over a semifield
admits a natural action of the monoid over that semifield associated with the
Kac-Moody datum and admits a cellular decomposition. This extends the previous
work of Lusztig, Postnikov, Rietsch and others on the totally nonnegative flag
manifolds (of finite type) and the work of Lusztig, Speyer, Williams on the
tropical flag manifolds (of finite type). As a by-product, we prove a
conjecture of Lusztig on the duality of totally nonnegative flag manifold of
finite type.
