More than half a million Rohingya refugees have flocked to Bangladesh since an army crackdown began on August 25 in Myanmar's Rakhine state, a process the UN has described as ethnic cleansing.
Shoaib Hossain Jewel, 25, and his 18-year-old Rohingya bride Rafiza have been on the run since marrying last month, said police in Jewel's home town of Singair.
"We heard he married a Rohingya woman. We went to his home at Charigram village to look for him," Singair police chief Khandaker Imam Hossain told AFP.
"But we did not find him there and his parents don't know where he has gone," he said, adding they were investigating the case.
In 2014 Dhaka banned marriages between Bangladeshis and Myanmar's Rohingya Muslim refugees following claims that members of the persecuted community were attempting to wed to gain citizenship in the mainly Muslim nation.
Jewel's father Babul Hossain said citizenship was not the motive this time and defended his son's marriage to Rafiza.
"If Bangladeshis can marry Christians and people of other religions, what's wrong in my son's marriage to a Rohingya?" Hossain told AFP.
"He married a Muslim who took shelter in Bangladesh."
The Dhaka Tribune newspaper said Jewel, a teacher in a madrassa or religious school, fell in love with Rafiza after her family fled the latest bout of violence in Myanmar and took refuge at a cleric's house in Singair.
In a police crackdown, the family was forced to move back to the main refugee camp in the southeastern district of Cox's Bazar -- some 265 miles from Singair.
Their wedding in Cox's Bazar was the first known one between a Bangladeshi and a Rohingya refugee since the August flare-up, the newspaper reported.
'Act Of Charity'
Many Bangladeshi men have travelled to the refugee camps since the influx began in hopes of marrying young Rohingya women, according to local media reports.
An AFP correspondent met a Bangladeshi man at Balukhali makeshift camp who came from a neighbouring village to find a bride for his elder brother.
"My brother wants to marry a Rohingya woman just to help her. He thinks marrying a girl from the distressed Muslim community will be treated as an act of charity," he said.
A senior police officer said they have stepped up surveillance in the camps to stop any such marriages and to combat trafficking of refugeegirls or children, many of whom fled to Bangladesh unaccompanied by parents.
"We are taking all preventive actions to ensure there are no marriages between Bangladeshis and Rohingya," Cox's Bazar's deputy police chief Mohammad Kazi Humayun Rashid told AFP.
He said authorities have banned any Bangladeshi or foreigner from entering the settlements after 5:00 pm.