  We propose stacked two-dimensional lattice designs of frustrated and SO(3)
symmetric spin models consisting of antiferromagnetic (AFM) triangular and
ferromagnetic (FM) sixfold symmetric sublattices that realize emergent Z3 Potts
nematic order. Considering bilinear-biquadratic spin interactions, our models
describe an SO(3)-symmetric triangular lattice AFM subject to a fluctuating
magnetization arising from the FM coupled sublattice. We focus on the classical
AFM-FM windmill model and map out the zero- and finite-temperature phase
diagram using Monte Carlo simulations and analytical calculations. We discover
a state with composite Potts nematic order above the ferrimagnetic
three-sublattice up-up-down ground state and relate it to Potts phases in
SO(3)-broken Heisenberg and Ising AFMs in external magnetic fields. Finally, we
show that the biquadratic exchange in our model is automatically induced by
thermal and quantum fluctuations in the purely bilinear Heisenberg model,
easing the requirements for realizing these lattice designs experimentally.
