In this paper we propose a globally stationary augmentation of the Exponential Smooth Transition Autoregressive (ESTAR) model that allows for multiple fixed points in the transition function. An F-type test statistic for the null of nonstationarity against such globally stationary nonlinear alternative is developed. The test statistic is based on the standard approximation of the nonlinear function under the null hypothesis by a Taylor series expansion. The model is applied to the U.S real interest rate data for which we find evidence of the new ESTAR process.