  Every expanding Thurston map $f$ without periodic critical points is known to
have an iterate $f^n$ which is the topological mating of two polynomials. This
has been examined by Kameyama and Meyer; the latter who has offered an explicit
construction for finding two polynomials in the unmating of the iterate.
Initializing this algorithm depends on an invariant Jordan curve through the
postcritical set of $f$--but we propose adjustments to this unmating algorithm
for the case where there exists a curve which is fully $f$-invariant up to
homotopy and not necessarily simple. When $f$ is a critically pre-periodic
expanding Thurston map, extending the algorithm to accommodate non-Jordan
curves in this manner allows us to unmate without iterates.
