  In previous work we showed that a loop $g\colon S^1 \to {\rm SU}(2)$ has a
triangular factorization if and only if the loop $g$ has a root subgroup
factorization. In this paper we present generalizations in which the unit disk
and its double, the sphere, are replaced by a based compact Riemann surface
with boundary, and its double. One ingredient is the theory of generalized
Fourier-Laurent expansions developed by Krichever and Novikov. We show that a
${\rm SU}(2)$ valued multiloop having an analogue of a root subgroup
factorization satisfies the condition that the multiloop, viewed as a
transition function, defines a semistable holomorphic ${\rm SL}(2,\mathbb C)$
bundle. Additionally, for such a multiloop, there is a corresponding
factorization for determinants associated to the spin Toeplitz operators
defined by the multiloop.
