This study develops a linear-rational multi-curve term structure model based on the Wishart affine process. The model allows for a stochastic correlation between the curves whilst the pricing of swaptions remains at par in terms of numerical complexity with caps and floors. We also show how the constant maturity swap (CMS) and the CMS spread option can be priced. We provide swaption and CMS spread option price approximations that are fast to evaluate and accurate. These approximations heavily rely on the affine property of the Wishart process. We illustrate how the model performs on real data by rolling a calibration using a 3-month long sample of at-the-money swaption data. We find that the estimated parameters are remarkably stable and the calibration procedure is robust. In particular, thanks to the specific Wishart properties the model can handle the stochastic correlation between the OIS term structure and the Euribor-OIS spread term structure.