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A causal Markov category with Kolmogorov products

In Fritz & Rischel, Infinite products and zero-one laws in categorical probability, the problem was posed of finding an interesting Markov category which is causal and has all (small) Kolmogorov products (there Problem 6.7). Here we give an example where the deterministic subcategory is the category of Stone spaces (i.e. the dual of the category of Boolean algebras) and the kernels correspond to a restricted class of Kleisli arrows for the Radon monad. We look at this from two perspectives. First via pro-completions and Stone spaces directly. Second via duality with Boolean and algebras and effect algebras.