It doesn't necessarily change the output distribution; it depends exactly how it's implemented, and Anthropic haven't told us that. Google's original SynthID paper describes how you can do this.

Toy proof-of-concept: Anthropic owns a secret key which is a coin-flip Bernoulli random variable K with p=1/2. You are paying Anthropic to give you X, a Bernoulli random variable with p=1/2. Anthropic changes from their old strategy, "draw from K, then throw it away and flip a coin, each time you ask for a sample", to their new strategy, "draw from K and send it to you". You cannot observe the difference, but Anthropic knows K and so they know when you are repeating its outputs. (Obviously this is a toy example; in reality the distribution is vastly more complicated than Bernoulli, and Anthropic isn't just storing some model outputs to use as K but instead is computing a correlation with a known pseudorandomness source.)