  In frustrated magnetic systems, the competition amongst interactions can
introduce extremely high degeneracy and prevent the system from readily
selecting a unique ground state. In such cases, the magnetic order is often
exquisitely sensitive to the balance among the interactions, allowing tuning
among novel magnetically ordered phases. In antlerite, Cu$_3$SO$_4$(OH)$_4$,
Cu$^{2+}$ ($S=1/2$) quantum spins populate three-leg zigzag ladders in a highly
frustrated quasi-one-dimensional structural motif. We demonstrate that at zero
applied field, in addition to its recently reported low-temperature phase of
coupled ferromagnetic and antiferromagnetic spin chains, this mineral hosts an
incommensurate helical+cycloidal state, an idle-spin state, and a multiple-$q$
phase which is the magnetic analog of misfit crystal structures. The
antiferromagnetic order on the central leg is reentrant. The high tunability of
the magnetism in antlerite makes it a particularly promising platform for
pursuing exotic magnetic order.
