In general, these are unidentifiable model with multiple modes. Usually, they are models with exchangeable parameters, mixture models with unconstrained parameters, there is rotation or normalization in the model etc. You can see another example here Inferencing trigonometric time series model
Informative prior is the answer, and the way to choose the right one is using prior predictive samples. You should have a look at [1708.07487] The prior can generally only be understood in the context of the likelihood
In practices, beside following good practice, lots trial and error is unavoidable. In the case of GP, it is known to be difficult to do inference using MCMC and you really need to have strong priors. You can have a look at Mike Betancourt’s Robust GP series: Robust Gaussian Processes in Stan