This paper develops a tractable model of a monetary union with a sound fiscal governance structure and shows how in such environment the design of monetary policy above and at the lower bound constraint on short-term interest rates can be linked to well-known findings from the literature dealing with single closed economies. The model adds a portfolio balance channel to a New Keynesian two-country model of a monetary union. If the monetary union is symmetric and the portfolio balance channel is not active, the model becomes isomorphic to the canonical New Keynesian three-equation economy in which central bank purchases of long-term debt (QE) at the lower bound are ineffective. If the portfolio balance channel is active, QE becomes effective and we prove that for sufficiently small shocks there exists an interest rate rule augmented by QE at the lower bound which replicates the equilibrium allocation and the welfare level of a hypothetically unconstrained economy. Shocks large enough to push the whole yield curve to the lower bound require, in addition, forward guidance. We generalise these results to an asymmetric monetary union and illustrate them through simulations, distinguishing between asymmetric shocks and asymmetric structures. In general, asymmetries give rise to current account imbalances which are, depending on the degree of financial integration, funded by private capital imports or through the central bank balance sheet channel. Moreover, our findings support that at the lower bound, as long as asymmetries between countries result from shocks, outcomes under an unconstrained policy rule can be replicated via a symmetric QE design. By contrast, asymmetric structures of the countries which matter for the transmission of monetary policy can translate into an asymmetric QE design.