The aim of this paper is to argue that the adoption of an unrestricted principle of bivalence is compatible with a metaphysics that (i) denies that the future is real, (ii) adopts nomological indeterminism and (iii) exploits a branching structure to provide a semantics for future contingent claims. To this end, we elaborate what we call Flow Fragmentalism, a view inspired by Kit Fine's non‐standard tense realism, according to which reality is divided up into maximally coherent collections of tensed facts. In this way, we show how to reconcile a genuinely A‐theoretic branching time model with the idea that there is a branch corresponding to the thin red line, that is, the branch that will turn out to be the actual future history of the world.