Abstract: We consider a mutation-finite quiver Q. Associated to Q are two interesting complex manifolds: (1) the space of stability conditions for the derived CY3 category of the Ginzburg algebra associated to the quiver, (2) the complex cluster Poisson variety. Each of these manifolds is constructed from the combinatorics of the quiver, though in very different ways. Tom Bridgeland and I introduce an "interpolation" of these complex manifolds: we give a construction of a complex manifold together with a submersion to the complex plane which we call the stability twistor space. The fiber over the origin is a finite quotient of the space of stability conditions whereas every other fiber is an etale cover of the cluster Poisson variety associated to the quiver. This etale map is a generalization of Thurston's grafting map. Constructing sections of the twistor family is closely related to solving the Riemann-Hilbert problem that encodes the Donaldson-Thomas invariants.