Germany will not recognise polygamy or marriages involving minors, Justice Minister Heiko Maas has said
"No-one who comes here has the right to put his cultural values or religious beliefs above our law," he told Bild.
In some Islamic countries, men are allowed to take up to four wives, but in Germany polygamy is banned by law.
Concern about polygamy and underage marriage has risen recently, as Germany has been taking in record numbers of migrants, many from Muslim countries.
The law in Germany is clear: no-one is allowed to be married to more than one person at a time, and that includes recent arrivals in the country.
But in practice polygamous relationships are often quietly tolerated, the paper reports. For example, if a man dies, his inheritance may be distributed between his two wives.
But Mr Maas wants to stop authorities turning a blind eye.
"Everybody must abide by the law, no matter whether he has grown up here or has only just arrived," he told the tabloid.
This also applies to forced and underage marriages.
"We cannot tolerate forced marriages, above all, if they affect under-age girls," Mr Maas said.
Bild reports that the federal state of Bavaria alone has registered 550 cases of brides aged under 18, and 161 under 16, among the asylum seekers that have arrived in the recent migrant wave.
In most of these cases, the young girls were already married before their arrival in Germany.
Syrian "child brides" and international law
- "Alarming increase" in number of child marriages within Syrian refugee communities in Jordan, Iraq and Lebanon, says Save the Children
- It is argued that marriage protects the girls from sexual assault and other hardship
- It is also seen as safeguarding family honour
- It reduces economic burden on refugee families
- But child marriage threatens a girl's physical and mental health
- It is outlawed by the Convention on the Elimination of All Forms of Discrimination against Women
- The right to free and full consent from both parties is enshrined in the Universal Declaration of Human Rights (article 16)
There is no clear law in Germany regarding what to do with underage marriages conducted in Muslim countries, and courts make judgements on a case-by-case basis.
A court in the city of Bamberg recently decided that the marriage of a 15-year-old girl to a 21-year-old man, conducted under Syrian jurisdiction, could be regarded as legal in Germany.
In Germany the minimum age for marriage is 18, but it is possible for a young person aged 16 or 17 to marry, if a family court gives consent, and the other party to the marriage is at least 18.
A note on terminology: The BBC uses the term migrant to refer to all people on the move who have yet to complete the legal process of claiming asylum. This group includes people fleeing war-torn countries such as Syria, who are likely to be granted refugee status, as well as people who are seeking jobs and better lives, who governments are likely to rule are economic migrants.