* Trump says "very productive" meetings with Chinese president
* Chinese state media says countries can manage their differences
* Issues like market access only partially addressed
* Industry sources see tough U.S. trade action on China ahead (Recasts, adds details)
BEIJING, Nov 10 (Reuters) - U.S. President Donald Trump lauded his Beijing meetings on trade and North Korea as "very productive," ending a visit which Chinese media declared set a "new blueprint" for handling ties, even as the White House looks set for tougher action on China.
Trump pressed China to do more to rein in North Korea's nuclear ambitions and said bilateral trade had been unfair to the United States, but also praised Xi's pledge that China would be more open to foreign firms.
Hours after Trump left Beijing on Friday, China said it would drop foreign ownership limits on local banks and asset management companies while loosening stake restrictions in securities firms and insurers - moves that have been long awaited by foreign financial firms.
Beijing faces intensifying pressure from Western governments and business lobbies to remove investment barriers and other rules that hobble overseas firms from operating in the country, as well as intellectual property theft.
Washington has refrained from pushing harder on trade because it needs China's cooperation on North Korea, although Xi, at least in public, went no further than reiterating China's determination to achieve denuclearisation through talks.
Trump and Xi, who call themselves friends, also oversaw the signing of about $250 billion in commercial deals, a show that some in the U.S. business community and others say detracts from addressing structural impediments that puts them at a disadvantage to Chinese rivals.
China may be mistaken in thinking it had done enough to address U.S. concerns, and Trump could find people disappointed at home he didn't make more progress on items like market access and North Korea, Paul Haenle, Director of the CarnegieTsinghua Center for Global Policy in Beijing, said.
"My concern is you may see a shift towards a much harder line coming from the U.S. administration. That will be a huge surprise to China and President Xi, especially given that Xi likely feels he has done a lot for Trump on this visit."
Trump reiterated in a tweet just before leaving Beijing for the APEC summit of Asia Pacific leaders in Vietnam that he didn't blame China for the trade gap between the two countries, adding that he had "very productive" meetings on trade and North Korea with Xi.
"I don't blame China, I blame the incompetence of past Admins for allowing China to take advantage of the U.S. on trade leading up to a point where the U.S. is losing $100's of billions. How can you blame China for taking advantage of people that had no clue? I would've done same!"
A U.S. industry source said the implication of Trump blaming his predecessors for the trade deficit is that Trump won't make the same mistake - a warning to Xi, not a capitulation.
The Trump administration did not bring hard trade policy items to the discussion with Xi because U.S. officials don't want to "argue over crumbs," the source added.
"Barring some dramatic action on North Korea by Xi, there is going to be a hard turn in U.S. trade policy," the source said.
A second source, who is close to the business delegation that traveled to Beijing this week, added that the Trump administration appeared poised to take a much harder line in the weeks ahead.
There were no obvious gaffes during the two-night stopover, and Trump and Xi seemed to enjoy being in each other's company.
At a banquet on Thursday night in the Great Hall of the People, Trump and Xi dined on coconut chicken soup, spicy chicken, stewed beef with tomatoes and grouper filets.
Chinese state media said the tone and outcome of Trump's visit had been largely positive, saying Trump and Xi were setting a new blueprint for handling relations and managing their differences.
"China has tried its utmost, even at the sacrifice of Sino-North Korean relations," influential tabloid the Global Times wrote in its editorial.
"Trump has gradually learned that Beijing is indeed making selfless contributions to promoting the denuclearisation of the peninsula. He can't demand more." (Reporting by Ben Blanchard; Additional reporting by Steve Holland and Matthew Miller, and Michael Martina in DANANG, Vietnam; Editing by Michael Perry, Tony Munroe and Nick Macfie)