when perfect correlation just means… perfect!

When looking at an X validated question on generating two perfectly negatively correlated Bernoulli variates last week, my intuition was that one had to be the opposite of the other, which means their parameters had to sum up to one. Intuition that was plain easy to back up by solving the equation

corr(C¹,C²)=-1

in terms of the joint distribution of (C¹,C²). That perfect correlation implies strong constraints on the parameter of the Bernoulli is not highly surprising given its binary support. Although I had no time to pursue the issue, I idly wondered at the generalisation to, say, a Binomial case, i.e., whether or not this case still is the only possible one for the above to hold. But again a perfect correlation can only occur with perfect prediction, i.e., when the Binomial variates have the same number of trials and complementary probability parameters. (Of no particular relevance is the fact that the originator of the question preferred an answer that showed how to simulate two Bernoulli such that C¹+C²=1!)