Real convergence among the ten EU 2004 accession economies is investigated with respect to long-run real interest parity. We employ a novel approach where unit-root tests for real interest differentials are embedded within a Markov regime-switching framework. Whereas standard univariate unit-root tests provide mixed support for parity, we find parity is present in all cases where differentials either switch between regimes of stationary and non-stationarity behaviour, or between alternative regimes of stationarity characterized by differing degrees of persistence. Further insights are obtained from the inferred probabilities of being in each regime, and the regime-switching nature of the differential variances.