matches well with experiments on [[Cs$_2$CuCl$_4$]{}]{}. The associated phase diagrams in the temperature-magnetic field plane are shown schematically in Fig. \[fig: magnetic\_phases\]. Details of each phase and its properties can be found in the appropriate section of the main text.

![Schematic phase diagrams in the temperature-magnetic field plane for fields along (a) the $a$ axis, (b) the $b$ axis, and (c) the $c$ axis. (d) Schematic phase diagram for the ideal 2d ($J$-$J'$) model. Here we use the abbreviations: FP = fully polarized state; AF = commensurate antiferromagnetic state; IC = incommensurate state; CAF = collinear antiferromagnetic state; and SDW = spin density wave state. The shaded areas denote the regions in which Dzyaloshinskii-Moriya and exchange non-trivially compete. For these regions, we do not have reliable theoretical predictions at this point.[]{data-label="fig: magnetic_phases"}](magnetic_phases.pdf){width="\columnwidth"}

![A single triangular layer of the cone state, illustrated for a field along the $a$ axis. Circles with arrows indicate the sense of precession of the spins, as one moves along the $x$ axis. This is most easily seen intuitively by comparing every other spin, which compensates for the natural staggering due to the underlying Néel correlations of the 1d chains. Note that in the cone state, all 