For the model given by some independent pairs $(x_i,y_i)$ identically generated from a bivariate Gaussian distribution, there is the convenient semi-conjugate family of "Normal-Wishart" prior distributions. It is mainly convenient because the posterior distributions are easy to simulate, without resorting to MCMC techniques. In particular the Jeffreys non-informative prior is at the boundary of the semi-cojugate family and the Jeffreys posterior is easily simulated.
But for the case when we assume equal variance for $x_i$ and $y_i$, is there a posterior distribution which is easy to simulate and achieving a good "frequentist performance" such as the Jeffreys posterior ? (that is, roughly speaking, the frequentist coverage of the $95\%$ credibility intervals approximately is $95\%$, for usual parameters of interest).