Japan's Nikkei fell 2.0%, resuming trade after a market holiday on Monday while MSCI's broadest index of Asia-Pacific shares outside Japan lost 0.2%.
MSCI's ACWI shed 0.13%, a day after the gauge of the world's 50 stock markets lost 1.63%, its biggest decline in two months, and leaving it teetering at its lowest level in two months.
Markets in mainland China and Taiwan are still closed for a public holiday on Tuesday for mid-Autumn festival while Korean markets remain shut through Wednesday.
On Wall Street, S&P500 shed 1.70%, the biggest drop in four months. Nasdaq Golden Dragon China index of U.S.-listed Chinese shares tumbled 5.4% to edge near 15-month lows hit last month.
Shares in Evergrande, which has been scrambling to raise funds to pay its many lenders, suppliers and investors, sank 10.2% on Monday.
Regulators have warned that its $305 billion of liabilities could spark broader risks to China's financial system if its debts are not stabilized.
"Whether Evergrande can make payments, and if not, whether the authorities will bail it out Those are the immediate questions," said Masahiro Ichikawa, chief strategist at Sumitomo Mitsui DS Asset Management.
Investors also fear troubles at China's property sector, one of the key engines of the country's stellar economic growth over decades, could have ripple effects across the global economy.
"In the longer term, we could see slower Chinese growth hurting surrounding countries," Ichikawa said.
In the currency market, the Chinese yuan weakened to 6.4830 per dollar.
The euro traded at $1.1726, after having touched a near-one-month low of $1.1700 while the safe-haven yen ticked up to 109.39 yen to the dollar.
The 10-year U.S. Treasuries yield dropped to 1.312%, though moves were relatively subdued as investors looked to the U.S. Federal Reserve's two-day policy meeting starting on Tuesday.
Investors are looking for the tapering timeline on its bond purchases as well as its board members' long-term rates and economic projections. This week will see policy decisions from many other central banks spanning Brazil, Britain, Hungary, Indonesia, Japan, Norway, the Philippines, South Africa, Sweden, Switzerland, Taiwan and Turkey.
Oil prices also rebounded a tad in Asia after falls the previous day. U.S. crude futures traded at $70.68 per barrel.
Waning risk appetite hit cryptocurrencies, with bitcoin slumping to 1 1/2-month low of $42,454.
(Additional reporting by Lewis Krauskopf in New York Editing by Shri Navaratnam) ((firstname.lastname@example.org; +81 3 4520 1195;))
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