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We analyze the phases of an S=1/2 spin model on a lattice of coupledtetrahedra. The presence of both Heisenberg and antisymmetric,Dzyaloshinsky-Moriya interactions can lead to two types of symmetry-brokenstates: non-magnetic dimer order and, unexpectedly, exotic 4 sub-lattice weakantiferromagnetic order - a state with a generically small ordered moment andnon-zero chirality. External magnetic field also induces weakantiferromagnetism co-existing with strong dimer correlations in the groundstate.  These states are formed as a result of broken Ising symmetries and exhibit anumber of unusual properties.Comment: 5 pages, 4 figures; final version to appear in Phys. Rev.

Weak antiferromagnetism and dimer order in quantum systems of coupled tetrahedra