Predicting a point-referenced spatial process at unobserved locations is one of the typical goals of a geostatistical analysis. If the spatial process is assumed to admit a covariance function, then to generate such predictions, it is necessary to know the form of the covariance function. In many instances, mostly because of computational convenience, researchers opt for a stationary covariance function, even though in reality the process might be non-stationary. In this paper, building upon the M-RA approach of Katzfuss (2017), we present a Bayesian hierarchical modeling framework that allows to handle both stationary data and globally non-stationary but locally stationary data, without the need to specify a priori a non-stationary covariance function. In both simulation experiments and a real data application, we show that our model, the mixture M-RA, obtained by embedding mixture priors within the M-RA framework, not only allows to detect regions of local stationarity but also outperforms other standard spatial modeling approaches.