* European shares climb after Asia sinks into the red
* Bond yields fall globally, Bunds at new low
* UK bond yields, pound under pressure
* Euro zone and UK PMI data weak
By Marc Jones
LONDON, July 3 (Reuters) - Investors returned to bonds on Wednesday, setting off another slide in benchmark debt yields, amid fears of a global trade war and recession, bets central banks will cut interest rates and falling oil prices.
European Union leaders' nomination of Christine Lagarde, the head of the International Monetary Fund, to replace Mario Draghi as president of the European Central Bank reinforced expectations of more monetary policy easing if it's needed.
Traders greeted the decision by sinking German 10-year Bund yields to record lows of minus 39 basis points, lowering Italian two-year yields back into negative territory for first time in over a year and lifting stocks and U.S. futures.
The yield on 10-year UK gilts fell 4 basis points to 0.687% , which left it below the Bank of England's main policy rate for the first time in a decade. U.S. Treasury yields slumped to their lowest since late 2016.
"We have already seen some weak data in recent weeks, so that is the backdrop," said Elwin de Groot, head of macro strategy at Rabobank. "And now have Christine Lagarde as the likely successor of Mr Draghi at the ECB, which for the market says that the dovish policies will continue."
European shares took little notice of some sizeable decline in Asia's stock markets and pushed 0.8% higher. Gains were led by an unusual pairing of defensive healthcare stocks and carmakers, which both jumped 1.2%.
There was plenty of data to digest, too. Euro area business activity picked up last month, figures showed, but remained weak. An upturn in services offset a downturn in factory output.
Worryingly, forward-looking indicators did not point to a rebound, and other data showed Britain's economy apparently shrank in the second quarter.
"The latest downturn has followed a gradual deterioration in demand over the past year as Brexit-related uncertainty has increasingly exacerbated the impact of a broader global economic slowdown," Chris Williamson, chief business economist at IHS Markit, said of the UK reading.
In the currency markets, the pound flirted with two-week lows and stood at $1.2568, on course for its fifth drop in the past six sessions.
The euro was steadier at $1.1282. The dollar dropped to 107.70 yen, off Monday's high of 108.535, which followed a weekend agreement between the United States and China to resume trade talks.
Sweden's crown reached a 2 1/2-month high of 10.4890 to the euro after the Riksbank said it remained on track to raise rates by early 2020, albeit with some caveats.
Oil prices also rose after data showed U.S. crude stockpiles fell more than expected last week. They remained wobbly, however, after falling more than 4% on Tuesday, even after OPEC and allies including Russia agreed to extend supply cuts.
Brent crude futures rose 0.7% to $62.85 per barrel. U.S. West Texas Intermediate crude futures gained 0.6% to $56.56 a barrel after dropping 4.8% the day before.
Gold gained 0.5% to $1,425.64 per ounce after rising as high as $1,435.99. (Reporting by Marc Jones; editing by Larry King)