Motivated by the recent discovery of the spin tube [(CuCl2tachH)3Cl]Cl2, we investigate the properties of a frustrated three-leg spin tube with antiferromagnetic intra- 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 whole gapped phase and slightly beyond. The implications for [(CuCl2tachH)3Cl]Cl2 are discussed, with the conclusion that this system behaves essentially as a spin-3∕2 chain.