This paper aims to find the time-consistent equilibrium strategy for a mean-variance portfolio selection problem under a non-Markovian regime-switching model, in which the coefficients are adapted to the filtration generated by a Markov chain. By introducing and investigating systems of coupled backward stochastic differential equations driven by the Markov chain, we obtain feedback representations of both open-loop equilibrium strategies and linear closed-loop equilibrium strategies. We also make further comparisons with the existing literature and reveal several interesting facts arising from the non-Markovian regime-switching model.