  Motivated by the recent discovery of the spin tube
[(CuCl$_2$tachH)$_3$Cl]Cl$_2$, we investigate the properties of a frustrated
three-leg spin tube with antiferromagnetic intra-ring and inter-ring couplings.
We pay special attention to the evolution of the properties from weak to strong
inter-ring coupling and show on the basis of extensive density matrix
renormalization group and exact diagonalization calculations that the system
undergoes a first-order phase transition between a dimerized gapped phase at
weak coupling that can be described by the usual spin-chirality model and a
gapless critical phase at strong coupling that can be described by an effective
spin-3/2 model. We also show that there is a magnetization plateau at 1/3 in
the gapped phase and slightly beyond. The implications for
[(CuCl$_2$tachH)$_3$Cl]Cl$_2$ are discussed, with the conclusion that this
system behaves essentially as a spin-3/2 chain.
