Sampling Stationary Vector Autoregression coefficients while Gibbs Sampling

I have been estimating a Bayesian Vector Autoregression using Gibbs Sampling. When constructing the posterior predictive distribution, I have noticed that when the simulated coefficients from the MCMC are not stationary (in the sense that the characteristic roots are not in the unit circle) then the forecasts tend to blow up.

I have tried to come up with some solutions to this issue that don't involve keeping a very tight prior. They seem to work, but are slow. For instance, I have tried iteratively sampling from the coefficients for each dependent variable and only moving on to the next if the maximum of the modulus of the eigenvalues is less than 1. Another approach I considered is to allow the burned simulations to potentially include non-stationary processes, but then the final output to only include stationary processes. I'm not sure if this is not a correct approach or not. Another approach I wasn't sure of was to replace the simulated coefficients with the posterior if it was stationary and the simulated ones were not.

I'm wondering if there are some reasonable approaches that aren't as slow?