Unanimity on the question of UK exit from the EU is not within reach, but this does not mean that the House of Commons or the population at large can not find a way out of the current Brexit impasse that is fair and legitimate. We discuss different voting procedures which satisfy some important principles of democracy and which can select the option that can win a majority against all other alternatives in a head-to-head majority vote. We argue that strategic considerations play an important role and we propose a procedure that works well and can help break the impasse when voters act strategically. The procedure requires (1) that all options with some minimum support are on the agenda, (2) that voting takes place in multiple rounds and (3) that in each round the alternative with the least support is eliminated until in the last round only two alternatives are left and the majority winner is selected. We discuss how this procedure can be modified to take into account that some voters may vote non-strategically and how it, in practice, could be used either in the House of Commons or in a new referendum.