  There is a correspondence between integrable lattice models of statistical
mechanics and discrete integrable equations which satisfy multidimensional
consistency, where the latter may be found in a quasi-classical expansion of
the former. This paper extends this correspondence to interaction-round-a-face
(IRF) models, resulting in a new formulation of the consistency-around-a-cube
(CAC) integrability condition applicable to five-point equations in the square
lattice. Multidimensional consistency for these equations is formulated as
consistency-around-a-face-centered-cube (CAFCC), which namely involves
satisfying an overdetermined system of fourteen five-point lattice equations
for eight unknown variables on the face-centered cubic unit cell. From the
quasi-classical limit of IRF models, which are constructed from the continuous
spin solutions of the star-triangle relations associated to the
Adler-Bobenko-Suris (ABS) list, fifteen sets of equations are obtained which
satisfy CAFCC.
