Let $X_{1,1},X_{1,2},X_{2,1},X_{2,2}$ be identically distributed r.v.s with distribution $\sim Be(p)$ and equally par-wise correlated, with pair-wise Pearson correlation coefficient $\rho$, e.g. $Corr[X_{1,1},X_{1,2}]=\rho$, $Corr[X_{1,1},X_{2,1}]=\rho$ and so on. We define $Y_1 = X_{1,1}\cup X_{1,2}$ and $Y_2 = X_{2,1}\cup X_{2,2}$, meaning that $Y_{1}=0$ if and only if $X_{1,1}=0,X_{1,2}=0$ and $Y_1=1$ otherwise. It follows that $Y_1$ and $Y_2$ are also Bernoulli distributed with probability $q$; and $q$ can be computed according to the discussion here, resulting in $q=\rho \cdot p (1-p)+p^2+2p(1-p)(1-\rho)$.
The question is how to calculate the correlation coefficient $\rho'$ between $Y_1$ and $Y_2$ as a function of $p$ and $\rho$.
Note: I estimate it using simulations, leading to $\rho'$ being somewhat larger than $\rho$, but I have not been able to find an analytical answer to this.
To compute $\rho'$ amounts to compute the full joint probability of $Y_1,Y_2$, in particular $P(Y_1=0,Y_2=0)$. This event corresponds to the four original rv being zero. But you cannot compute this because you only have the first and second moments, and this alone does not determine the joint distribution of the four original variables.