The best unbiased estimate of a Bernoulli variance

You have independent flips $X_1, \dots, X_n$ from a Bernoulli$(p)$ and want the best unbiased estimate of $\theta = p(1 - p)$, the variance of a single flip.

Starting from the crude estimator $X_1(1 - X_2)$, build the minimum-variance unbiased estimator by conditioning on the sufficient total $T = \sum_i X_i$.