Before revision:  We introduce an ancestral process that keeps track of the sampling distribution along the ancestral structures and that satisfies a duality relation with the type-frequency process. We refer to it as Bernstein coefficient process and to the relation as Bernstein duality. The latter is a generalisation of the classic moment duality. Many classic results in the restricted setting of a moment duality generalise into our framework. In particular, we derive criteria for the accessibility of the boundary and determine the time to absorption.
After revision:  Building on this object, we construct a continuous-time Markov chain and relate it to the forward process via a new form of duality, which we call Bernstein duality. We adapt classical methods based on the moment duality to determine the time to absorption and criteria for the accessibility of the boundary and determine the time to absorption.