  We investigate an antiferromagnetic triangular Ising model with anisotropic
ferromagnetic interactions between next-nearest neighbours, originally proposed
by Kitatani and Oguchi (J. Phys. Soc. Japan {\bf 57}, 1344 (1988)). The phase
diagram as a function of temperature and the ratio between first- and second-
neighbour interaction strengths is thoroughly examined. We search for a
Kosterlitz-Thouless transition to a state with algebraic decay of correlations,
calculating the correlation lengths on strips of width up to 15 sites by
transfer-matrix methods. Phenomenological renormalization, conformal invariance
arguments, the Roomany-Wyld approximation and a direct analysis of the scaled
mass gaps are used. Our results provide limited evidence that a
Kosterlitz-Thouless phase is present. Alternative scenarios are discussed.
