  In the context of the Franks-Misiurewicz Conjecture, we study homeomorphisms
of the two-torus semiconjugate to an irrational rotation of the circle. As a
special case, this conjecture asserts uniqueness of the rotation vector in this
class of systems. We first characterise these maps by the existence of an
invariant foliation by essential annular continua (essential subcontinua of the
torus whose complement is an open annulus) which are permuted with irrational
combinatorics. This result places the considered class close to skew products
over irrational rotations. Generalising a well-known result of M. Herman on
forced circle homeomorphisms, we provide a criterion, in terms of topological
properties of the annular continua, for the uniqueness of the rotation vector.
As a byproduct, we obtain a simple proof for the uniqueness of the rotation
vector on decomposable invariant annular continua with empty interior. In
addition, we collect a number of observations on the topology and rotation
intervals of invariant annular continua with empty interior.
