A presidential election between Donald Trump and Bernie Sanders — still a fairly unlikely prospect but, well, never say never — might seem like the perfect opportunity for an independent candidate to enter the race. Sanders would be the most left-wing nominee since George McGovern, who lost by 23 points to Richard Nixon in 1972. Trump, who has a net favorable rating of -25 percentage points among general election voters, would begin the race as perhaps the most unpopular major-party nominee ever.

It wouldn’t be as easy as it sounds for an independent, though. That’s partly because, in the conditional probability warp zone I’m asking us to consider, Sanders and Trump will each have won their respective nominations and demonstrated some political acumen in the process. They might be below-average candidates, but they wouldn’t be total pushovers. It’s also because, as Brendan Nyhan reminds us, there are some significant structural challenges facing an independent candidate. Among other things, a centrist, independent candidate would probably underperform a bit in the Electoral College relative to his or her share of the popular vote.

But neither of those problems are quite what I had in mind. Instead, there’s a more basic issue: It’s hard to find the right candidate for the job.

Take former New York Mayor Michael Bloomberg, who has been floating trial balloons about an independent bid and who seems more likely to run in the event of a Trump vs. Sanders matchup. Bloomberg, worth an estimated $49 billion, would have no trouble financing a presidential campaign. Surely he could find some wiggle room between Trump and Sanders?

Maybe not. Sanders, obviously, would be an extremely liberal Democratic nominee. Trump, perhaps less obviously, might wind up being fairly left-of-center for a Republican candidate. So the center-left Bloomberg would be swimming in a crowded lane.

We can visualize this using data from OnTheIssues.org, which categorizes the ideological position of politicians based on their public statements and voting records. OnTheIssues helpfully distinguishes between social and economic issues, allowing for candidates to be “populist” (economically liberal and socially conservative) or “libertarian” (economically conservative and socially liberal) rather than simply conventionally liberal or conservative. And because OnTheIssues has been around for some time, we can track how a candidate’s positions have “evolved” with the political winds. The chart below shows the scores for Trump, Sanders and Bloomberg — both where they stand now and how they positioned themselves earlier in their careers.

Today’s Donald Trump is often described as a “populist.” But if you define “populism” as OnTheIssues does, meaning someone who’s socially conservative but economically liberal, that was more true of him back in 1999 and 2000. That’s when Trump was considering his own independent bid for president, calling for a wealth tax on multi-millionaires while already emphasizing the importance of America “[controlling] its own borders.” Recently, Trump’s positions have become more conservative, although they remain fairly idiosyncratic. Where he’d wind up as a general-election candidate is anyone’s guess, but it would probably be somewhere along the spectrum between populist and moderately conservative.

Sanders has also “evolved,” although less dramatically than Trump. He had a few traces of crusty, Vermont populism before, with relatively moderate positions on immigration and guns, but has gradually shed them to adopt more conventionally liberal positions. In fact, he’s become a sort of liberal lodestar: Sanders now rates as maximally liberal on both economic and social policy, according to OnTheIssues.

Bloomberg, who was registered as a Republican when he ran for mayor in 2001 and 2005 but became an independent in 2007, has also moved to his left and now holds positions that, while rightward of Sanders’s, aren’t appreciably different from Barack Obama’s or Hillary Clinton’s. In the event of a presidential bid, Bloomberg would presumably pivot back toward the center. He used to be a Democrat, but even in his days as a nominal Republican he was really a center-left moderate. And because of his paternalist streak on everything from guns to Big Gulps, Bloomberg can’t be really be described as a “libertarian.”

So while a Sanders-Trump-Bloomberg election would leave voters on the center-left with several plausible choices, other groups of voters would be neglected. Notably, there wouldn’t be a true, Ronald Reagan-style conservative in the race. The election would also be something of a nightmare for libertarian-inclined voters forced to consider Sanders’s big government, Trump’s “yuge” government and Bloomberg’s technocracy.

You might also notice some demographic similarities between Bloomberg, Sanders and Trump, all old loudmouthed white guys from New York. That’s not exactly a diverse group in an election that had once seemed like it might produce the first female or Hispanic president.

Part of the challenge for everyone else is that Trump is more like a third-party candidate than a Republican in many respects, perhaps the closest thing to Ross Perot we’ve seen for a long time. Judging by his high unfavorable ratings, Trump’s appeal is not especially broad, but it cuts through the political coalitions we’re used to at some odd angles.

If you had to choose, though, an independent Mitt Romney-esque candidate on the center-right might seem to have a better chance in a Trump-Sanders race than Bloomberg on the center-left. Such a candidate could build a coalition from well-educated Republicans in the suburbs, who are not very inclined toward Trump and make up about 21 percent of the electorate, according to our swing-the-election interactive. Nonwhite Republicans might also be a pretty good fit for this hypothetical candidate, although they’re only 6 percent of the electorate.

The projected 2016 electorate GROUP DEMOCRATIC REPUBLICAN College-educated white 16% 21% Non-college-educated white 13% 21% Black, Hispanic, Asian and other 24% 6%

Where would such a candidate go from there? That’s tricky. White Republicans without college degrees are likely to be Trump’s best group. College-educated white Democrats are pretty into Sanders. Could this hypothetical candidate draw some votes from blacks and Hispanics who usually vote Democratic? Maybe if the candidate were black or Hispanic. So, uhh … Nevada Gov. Brian Sandoval? Former Secretary of State Condoleezza Rice?

If those names seem ridiculous — Sandoval has little national name recognition, Rice has never run for office before and neither have anywhere near the money they’d need to mount a campaign — try coming up with your own list. It’s not easy. A two-party system in a time of intense partisanship doesn’t produce many truly independent-minded politicians or teach them how to extend their appeal beyond their bases. Nor are the candidates whom wonks and journalists find appealing necessarily the ones who might catch fire in Ohio. Trump, like him or not, might be the closest thing we’ve seen to a viable third-party candidate in a long time.