Let's begin from the de Finetti–Hewitt–Savage theorem: for an exchangeable sequence of random variables we can always write $$ p(x_1, x_2,\cdots) = \int \prod p(x_i | L) P(dL) $$ where $L$ is a latent variable upon which the variables become conditionally independent. I've seen it applied to exchangeable but correlated Gaussian variables - they become independent when conditioned upon another Gaussian variable.
Now let's take a symmetric Dirichlet distribution, i.e. a Dirichlet with concentration parameters $\alpha_i = \alpha$ for all $i$. Thus the variables are exchangeable. This distribution only has support over $\sum x_i = 1$, which means that the $x_i$ are not independent.
I think I must be making a mistake because the theorem seems too good to be true, as it implies that there is a latent variable under which the $x_i$ are conditionally independent. But I just can't see how there could exist a single latent variable to enforce the fact that it only has support over $\sum x_i = 1$.
Maybe I am just missing a simple assumption for the theorem? or maybe I am just not seeing something.