Suppose we have a Beta-binomial random variable $Z \sim \text{BetaBinomial}(\alpha, \beta, n)$. It satisfies \begin{align*} \mathbb{E}[Z] = n\frac{\alpha}{\alpha + \beta}, \qquad \text{Var}(Z) = n \frac{\alpha\beta}{(\alpha+\beta)^2}\frac{\alpha+\beta+n}{\alpha+\beta+1} \end{align*} Upon the reparametrization $\pi = \frac{\alpha}{\alpha + \beta}$ and $\rho = \frac{1}{\alpha + \beta + 1}$, \begin{align*} \mathbb{E}[Z] = n\pi, \qquad \text{Var}(Z) = n \pi(1-\pi)(1 + (n-1)\rho) \end{align*} We see that if we take $\mathbf{Y} = (Y_1, \cdots, Y_n) \in \{0, 1\}^n$ as a multivariate Bernoulli random variable with $\mathbb{E}[Y_i] = \pi$ and $\text{Corr}(Y_i, Y_j) = \rho$, we see that $Z$ and $\mathbf{1}^\intercal \mathbf{Y}$ have the same first and second moments.
But, the multivariate distribution of $\mathbf{Y}$ isn't defined by mean and pairwise correlations of its components. What specific distribution must be chosen for $\mathbf{Y}$ so that $\mathbf{1}^\intercal \mathbf{Y} \overset{\mathcal{D}}{=} Z$ (that is, the two are equal in distribution)?