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.
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