Based on a conservative Allen-Cahn phase field method, a three-dimensional nonorthogonal multiple-relaxation-time (MRT) lattice Boltzmann (LB) model for interface tracking in multiphase flow is proposed in this paper. Different from the traditional MRT LB model, the transformation matrix in the present model is constructed based on a set of nonorthogonal basis vectors to simplify the transformation process between the discrete velocity space and the moment space. Therefore, a higher computational efficiency is achieved by the present model. The present model is developed on two different three-dimensional lattice sets (D3Q19 and D3Q27) to obtain a thorough perspective about the performance of the nonorthogonal matrix. Coupled with the nonorthogonal transformation matrix, simplified discrete source terms are also developed for both two lattice sets to further improve the efficiency of the present model. Numerical tests demonstrate that compared with the traditional MRT LB model, the present model shows a significantly higher computational efficiency and better stability while maintaining a comparable accuracy. It is also found that the D3Q19 nonorthogonal model does not obviously weaken the accuracy of D3Q27 nonorthogonal model while D3Q27 nonorthogonal model dose not decrease the stability of the D3Q19 nonorthogonal model, which is different from the orthogonal model.