  We show that the off-diagonal exchange anisotropy drives Mott insulators with
strong spin-orbit coupling to a classical spin liquid regime, characterized by
an infinite number of ground states and Ising variables living on closed or
open strings. Depending on the sign of the anisotropy, quantum fluctuations
either fail to lift the degeneracy down to very low temperatures, or select
non-collinear magnetic states with unconventional spin correlations. The
results apply to all 2D and 3D tri-coordinated materials with bond-directional
anisotropy, and provide a consistent interpretation of the suppression of the
x-ray magnetic circular dichroism signal reported recently for
$\beta$-Li$_2$IrO$_3$ under pressure.
