By Wayne Cole

SYDNEY, March 31 (Reuters) - Asian shares managed a tentative rally on Tuesday as factory data from China held out the hope of a rebound in activity even as other countries across the globe all but shut down.

China's official manufacturing purchasing managers' index (PMI) bounced to 52.0 in March, up from a record-low 35.7 in February and topping forecasts of 45.0.

Analysts cautioned the index could overstate the true improvement as it measures the net balance of firms reporting an expansion or contraction in activity.

If a company merely resumed working after a forced stoppage, it would read as an expansion without saying much about the overall level of activity.

Still, the headline number was a relief and helped MSCI's broadest index of Asia-Pacific shares outside Japan rise 1.1%.

Japan's Nikkei firmed 1.0% after a jittery start, while South Korea added 2%.

E-Mini futures for the S&P 500 added another 0.6%, supported by talk of book-keeping demand.

"It's month-end rebalancing, whereby balanced funds now underweight equities versus fixed income given this month's valuation destruction, need to buy stocks to get back into balance," analysts at NAB said.

Healthcare had led Wall Street higher, with the Dow ending Monday up 3.19%, while the S&P 500 gained 3.35% and the Nasdaq 3.62%.

News on the coronavirus remained grim but radical stimulus steps by governments and central banks have at least provided some comfort to economies.

Infections in hard-hit Italy slowed a little, but the government still extended its lockdown to mid-April. California reported a steep rise in people being hospitalized, while Washington state told people to stay at home.

Trade ministers from the Group of 20 major economies agreed on Monday to keep their markets open and ensure the flow of vital medical supplies.

OIL PRICES OVERWHELMED

Portfolio management also played a part in the forex market where many fund managers found themselves over-hedged on their U.S. equity holdings given the sharp fall in values seen this month, leading them to buy back dollars.

That saw the euro ease back to $1.1020, from a top of $1.143 on Monday, while the dollar index bounced to 99.330, from a trough of 98.330.

Month-end demand for dollars from Japanese funds saw the dollar inch up to 108.45 yen, though it remained some way from last week's peak at 111.71.

Oil prices plunged to the lowest in almost 18 years on Monday as lockdowns for the virus squeezed demand even as Saudi Arabia and Russia vied to pump more product.

In a new twist, U.S. President Donald Trump and Russian President Vladimir Putin agreed during a phone call on Monday to have their top energy officials meet to discuss slumping prices.

"However, the reality is that the level damage to demand is likely to overwhelm any production cut agreement between major producers," wrote analysts at ANZ in a note.

"The lockdown of cities around the world and the shutdown of the aviation industry will cause a fall in demand the industry has never seen before."

Prices did at least try and steady early Tuesday, with U.S. crude up 56 cents to $20.64. Brent crude futures gained 19 cents to $22.95 a barrel.

In the gold market all the talk has been of a rush of demand for the physical product amid shortages in coins and small bars. Flows into gold-backed ETFs have ballooned by $13 billion so far this year, the most since 2004.

The metal was holding at $1,615 an ounce, well up from a low of $1,450 touched early in the month.

(Editing by Sam Holmes & Shri Navaratnam)