A core challenge in the widespread adoption of structural health monitoring (SHM) is the generation of robust, damage sensitive features. Many damage detection algorithms implicitly assume that features are stationary in time, interpreting non-stationarity as evidence for damage. It is well understood in the SHM literature that non-damage effects such as environmental and operating variations (EOVs) can lead to non-stationarity in SHM features. Cointegration is a method that has been proposed as a solution to EOVs in SHM that projects several features onto a reduced basis wherein they are stationary in time. Applications of cointegration to SHM thus far have focussed on maximum-likelihood solutions for the cointegrating vectors; although effective, deterministic approaches are unable to account for uncertainties in the projection. In this work, a Bayesian view of cointegration is taken in the context of SHM for the first time. Because the underlying inference task is intractable for most forms of prior belief in the cointegrating vectors, a convenient Hamiltonian Monte-Carlo (HMC) within Gibbs sampling scheme is proposed. Access to distributional estimates of damage-sensitive features enables a robust novelty detection that correctly incorporates uncertainties arising from the cointegrating projection. The proposed approach is demonstrated on an experimental case study and found to give rise to highly robust damage sensitive features.