Whole-economy scenarios for limiting global warming to 1.5C suggest that direct carbon emissions in the buildings sector should decrease to almost zero by 2050, but leave unanswered the question how this could be achieved by real-world policies. We take a modelling-based approach for simulating which policy measures could induce an almost-complete decarbonisation of residential heating, the by far largest source of direct emissions in residential buildings. Under which assumptions is it possible, and how long would it take? Policy effectiveness highly depends on behavioural decision- making by households, especially in a context of deep decarbonisation and rapid transformation. We therefore use the non-equilibrium bottom-up model FTT:Heat to simulate policies for a transition towards low-carbon heating in a context of inertia and bounded rationality, focusing on the uptake of heating technologies. Results indicate that the near-zero decarbonisation is achievable by 2050, but requires substantial policy efforts. Policy mixes are projected to be more effective and robust for driving the market of efficient low-carbon technologies, compared to the reliance on a carbon tax as the only policy instrument. In combination with subsidies for renewables, near-complete decarbonisation could be achieved with a residential carbon tax of 50-200Euro/tCO2. The policy-induced technology transition would increase average heating costs faced by households initially, but could also lead to cost reductions in most world regions in the medium term. Model projections illustrate the uncertainty that is attached to household behaviour for prematurely replacing heating systems.