Can we test inconsistent empirical theories?

Despite the logical possibility and alleged existence of inconsistent empirical theories, the problem of their testability remains largely unexplored. I develop a testability criterion which makes some of these theories testable, including some observationally inconsistent ones. However, they are not rejectable qua inconsistent by this criterion. These results, while opening the domain of scientific theories to inconsistent ones, challenge the prospects of dialetheism in the philosophy of science.