  We prove a global Birkhoff decomposition for almost split real forms of loop
groups, when an underlying finite dimensional Lie group is compact. Among
applications, this shows that the dressing action - by the whole subgroup of
loops which extend holomorphically to the exterior disc - on the $U$-hierarchy
of the ZS-AKNS systems, on curved flats and on various other integrable
systems, is global for compact cases. It also implies a global infinite
dimensional Weierstrass-type representation for Lorentzian harmonic maps (1+1
wave maps) from surfaces into compact symmetric spaces. An "Iwasawa-type"
decomposition of the same type of real form, with respect to a fixed point
subgroup of an involution of the second kind, is also proved, and an
application given.
