  The 1-in-3 and Not-All-Equal satisfiability problems for Boolean CNF formulas
are two well-known NP-hard problems. In contrast, the promise 1-in-3 vs.
Not-All-Equal problem can be solved in polynomial time. In the present work, we
investigate this constraint satisfaction problem in a regime where the promise
is weakened from either side by a rainbow-free structure, and establish a
complexity dichotomy for the resulting class of computational problems.
