What would be the interpretation of asymmetric kernels in Gaussian Process Regression? This paper involves with asymmetric Kernels. They claim  that this arises due to local parameters. But this is not really true. They induce a particular asymmetric structure in the Kernel yet still call these "Gaussian processes".
What is an interpretation of this that makes sense? It isn't really a Gaussian process any more: it seems like one interpretation is that they are inducing some causal graph (directed instead of undirected random field) that gives rise to this asymmetric structure. This is just a guess.
Is there any better way to see how this is still somehow a "Gaussian Process" even thought the kernel (covariance function) is not symmetric?

Not an expert, but to my understanding their approach would break the PSD constraint on the corresponding covariance matrix. They don't address this in their paper at all. Like you mentioned, they call it a GP, but all they are using from the GP is the weighted sum portion that pertains to the mean. They ignore the covariance completely.
