With two byelections coming in Conservative seats, Labour should be happy. Byelection campaigns against a government mired in scandal are normally a dream come true for the opposition: a chance to mobilise discontent and demonstrate its own electoral credibility. While the two forthcoming contests – in Old Bexley and Sidcup and North Shropshire – are both in safe Conservative seats, safer seats than these fell in the 1990s, when opposition parties profited from simmering discontent with a long-running Tory government. Perhaps that time can come again. And yet and yet. Despite serial government scandals and blunders, and a once-in-a-century health crisis, Labour is still trailing in the polls. Voters are not happy with the government yet complain they do not know what the party opposing it would do differently. Keir Starmer is more popular than his predecessor but remains less popular than past opposition leaders who have gone on to win. Boris Johnson is untrusted, unpredictable and unpopular, yet voters still prefer him to his Labour rival when forced to choose. Labour, it seems, needs a bolder, clearer, more ambitious vision to take to the electorate. The problem with this theory is that it ignores the brute reality of the electoral map Labour must fight on. The past two elections have reshaped the political battle lines. The Conservatives’ gains in both 2017 and 2019 have come by uniting Brexit backers, a strategy that can deliver majorities because, while the Leave vote is not drastically larger than the Remain vote, it is much more efficiently spread across seats. Remain voters are concentrated in big-city seats; Leave voters are spread across the country. As a result, while only a narrow majority of voters backed Brexit, a large majority of constituencies have Leave majorities. A party that can win 80% of the Leave vote, as the Conservatives did in 2019, is well on the way to victory in most of these seats. While Brexit opened up a new route to victory for the Conservatives, it closed the jaws of an electoral trap for Labour. For years, a divide in outlook and values has been opening up between Labour’s past and present voters. Brexit drastically worsened this problem by placing the two groups on opposite sides of a tribal divide – the diverse youthful, university-educated, big-city dwellers who are becoming Labour’s new core vote became Remainers, while the ageing, socially conservative, small-town residents drifting away from the party became Leavers. Yet, while the Conservatives can win by focusing on one side of this divide, Labour cannot. The Remain vote is concentrated in too few seats and the Leave vote is too large and too evenly spread. Polarisation is a winning strategy for Conservatives. Bridging divides is the only winning strategy for Labour. This is politics on hard mode. To see just how hard it is, recall how attempts to bridge the Brexit divide defined and defeated Starmer’s predecessor. Many of Jeremy Corbyn’s most senior advisers saw the problems the new electoral map posed for their party and urged their leader to make a credible offer to the Leave voters who dominated the electoral battleground. But this argument never convinced Labour Remainers, who instead demanded a second referendum and began defecting to other Remain parties when Labour failed to deliver. Corbyn was trapped. He could not ignore Labour activists’ demands for a “people’s vote”, yet acceding fully to Remainers’ demands would commit Labour to a policy that was toxic in the Leave-leaning seats that would decide the election. Corbyn tried to square this circle by offering something for everyone – a new, Labour-flavoured Brexit deal for Leave supporters and a referendum on that deal for Remainers. But this attempt to appease both sides proved credible to neither. Corbyn’s compromise failed badly, but a more decisive stance might have been even worse. Labour lost equal numbers of Leavers and Remainers in 2019, but the Leave defections hurt more because they were congregated in marginal seats. Labour needed a stronger offer to compete with “get Brexit done” in these seats, yet a fuller embrace of Brexit would have risked an exodus in the party’s Remain heartlands. Although the Brexit crisis is now in a less acute phase, Starmer faces the same electoral dilemmas. Voters still divide into Leave and Remain tribes, these identities are still more widely shared than loyalties to any political party and Leave support is still better spread across seats. Labour cannot hope to win without finding a way to engage the Leave tribe while not alienating the Remain tribe. Can it be done? There are some reasons for hope. Issues that unite Leave and Remain voters – the economy, the NHS, public services – are higher on the agenda now, while immigration, the issue that polarises them most, has less resonance than at any point for a generation. Johnson’s magnetic attraction to tribal Leave voters disguises broader weaknesses – he has never been popular or trusted and his already weak ratings are in decline. The current Labour leader is less divisive than his predecessor and thus better able to put his case without having his message derailed by past or present controversies. Voters are more volatile than ever and decide who to back later. Labour will take heart from the recent German election, where an unfancied centre-left moderate rode a late polling surge to victory. While recent scandals will put a spring in Labour’s step, it cannot pin its hopes solely on a change in the public mood. The Conservatives know that polarisation remains the best route to victory on a Leave-leaning map and will keep seeking new “culture war” controversies to reopen the old Brexit divide. They have an unlikely ally in such efforts – committed progressives who want Labour to focus on an uncompromising socially liberal message and are as eager for divisive arguments over values as the Conservatives. Many of Labour’s own culture warriors do not see the risks polarisation poses for their party and see Starmer’s efforts to woo more socially conservative voters as an erosion of progressive values. Purity is always appealing to true believers, but purity cannot win Labour an election on today’s map. To win again, Starmer must instead help a divided party and a polarised nation learn to love compromise. This article was amended on 7 November 2021. One of the upcoming byelections is in North Shropshire, not North Somerset as stated in an earlier version. Dr Robert Ford is professor of political science at the University of Manchester and co-author of The British General Election of 2019