  We consider infinite or periodic 2D triangular Ising lattices with arbitrary
positive or negative nearest-neighbor couplings $K_i(\vec{r})$, where $\vec{r}$
and $i$ indicate the bond position and orientation, respectively. Iterative
application of the star-triangle transformation to an initial lattice
$\mathcal{T}(0)$ with a set of couplings $\{K_i^{(0)}(\vec{r})\}$ generates a
sequence of lattices $\mathcal{T}(n)$, for $n=1,2,\ldots,$ with couplings
$\{K_i^{(n)}(\vec{r})\}$. When $\mathcal{T}(0)$ includes sufficiently strongly
frustrated plaquettes, complex couplings will appear. We show that,
nevertheless, the variables $1/\sinh 2K_i^{(n)}\!(\vec{r})$ remain confined to
the union ${\mathbb{R}} \cup \rm{i}{\mathbb{R}}$ of the real and the imaginary
axis. The same holds for a lattice with free boundaries, provided we
distinguish between "receding" and "advancing" boundaries, the latter having
degrees of freedom that must be fixed by an appropriately chosen protocol. This
study establishes a framework for future analytic and numerical work on such
frustrated Ising lattices.
