WHEN President Nicolas Sarkozy was mulling over a reshuffle of his government last autumn and thinking of replacing his prime minister, one name that almost never surfaced was that of Christine Lagarde. To outsiders, this might seem surprising. France's first female finance minister, and its longest-serving since 1974, has made a name in international circles as a widely respected and skilled professional. But there is something about her straight-talking, business-like approach that has never been fully appreciated by the score-settling, rumour-mongering world of French politics. It took her growing reputation abroad to finally win her respect at home.

This mix of Frenchness and internationalism is very much Ms Lagarde's hallmark, and can be a source of tension. When she stepped off the plane from Chicago, where she was global president of Baker & McKenzie, a law firm, to join the government under President Jacques Chirac in 2005, almost her first public comment was to criticise the rigidity of France's labour market. The French were horrified; Ms Lagarde carefully rephrased her thoughts, and an early lesson was learned about how far it is possible to bring economically liberal ideas into the French political debate.

Indeed, appointed finance minister in 2007 by Mr Sarkozy, Ms Lagarde has not hesitated to defend what she considers to be in French interests, such as the regulation of hedge funds or an international tax on financial transactions. This leads some commentators to wonder what her real convictions are, and whether she has sacrificed her more market-friendly instincts in order to forge a political career in statist France. Yet she has also quietly got on with some liberalising reforms. While dutifully pushing some of Mr Sarkozy's dafter ideas abroad, she has also done a fair amount to try to inject more competition into the French economy (such as strengthening the anti-trust watchdog), and to boost public-sector efficiency (such as merging the job-placement and unemployment-benefit agencies).

Ms Lagarde's main strengths are a mix of hard-working professionalism, an appetite for technical detail, and an ability to get her way with charm rather than bullying. She often seems more at ease at global summits than on the benches of the unruly French National Assembly. It is hard to find anybody who has worked for her in France over the years who has a bad word to say about her. As a teenager, she was a member of the French national synchronised swimming team, and she likes to joke that this taught her not only teamwork and self-discipline, but also how to hold her breath. Her quick wit, in fluent English as well as French, even managed to win over Jon Stewart when she appeared on the Daily Show, bearing a French beret as a gift. In the past she has said that there was “too much testosterone” in high-powered circles, a comment that now looks prescient.

At a news conference on May 25th, Ms Lagarde conceded that being European was not necessarily an asset for a candidate to run the IMF, but hoped that it would not be regarded as a handicap either. Her central role in dealing with the euro-zone debt crises, during which she has been a consistent advocate of bailing out debt-ridden governments, could put her in a potentially awkward position at the fund. More worrying for the French, there is an outstanding legal inquiry hanging over her.

A public prosecutor is investigating whether there is any ground for a full inquiry into a decision Ms Lagarde made as finance minister linked to a long-running damages case brought against the state by Bernard Tapie, a French tycoon. She ruled that it should be taken out of the courts and settled through an arbitration panel. As a result, Mr Tapie was awarded more than €200m ($280m) in damages. Ms Lagarde said today that she has a “clear conscience” over the referral. Her advisers say that she followed procedure, and was acting in taxpayers' interests, as the court case was costing public money every year that it went unsettled. But the timing, in the wake of the arrest of Dominique Strauss-Kahn, is unfortunate. Right now, the French can ill-afford even the whiff of impropriety.