We study products of multitwists along filling multicurves on infinite-type surfaces and the geometry of their mapping tori. Unlike the finite-type case, the absence of invariant curves or arcs does not characterize hyperbolicity. However, under suitable boundedness assumptions, sufficiently large twisting produces hyperbolic mapping tori, including explicit families with unique hyperbolic metrics and bounded geometry. We are also able to recover Cannon-Thurston maps and geometric models through the SOL metric. However, the “dual” picture on quadratic differentials and measured lamination behaves quite differently. This is joint work in progress with Juan Souto and Ferrán Valdez.