All of Constance Marten and Mark Gordon's first four children were taken into care, a court has been told.
The Old Bailey heard the children were removed when a judge found there had been an incident of domestic violence.
The jury was also told police were called when the couple went to a hospital to have their first child in 2017, where they gave false names.
They deny manslaughter charges over their fifth child, baby Victoria, whose body was found in a shed.
In agreed facts read to the jury, the London court heard that Ms Marten first went to a hospital in London in June 2017 where she told them she had been living in a camper van.
Social services issued a "national hospital alert" to other parts of the NHS, which is issued when it is suspected a pregnant woman needs protection or support.
She then attended a hospital in Wales with Mr Gordon that winter where she was in the early stages of labour and admitted as an "unbooked case" - where the mother has had no NHS antenatal care.
She said her name was Isabella O'Brien, while Mr Gordon said his name was James Amer. Speaking with a fake Irish accent, she told staff she was from a travelling family, had never been to school and did not have an NHS number.
The jury heard previously that Ms Marten was from a wealthy family and by December 2022 was receiving a monthly allowance of £3,400.
When the couple's real identities were ascertained the police were called, as well as social services.
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She said they had moved to Wales to get away from her family.
It emerged that the couple had been living in a tent in a wooded area and had only bought some clothes and nappies in preparation for the arrival of their first child.
Ms Marten accompanied a social worker to the tent, which was described as a "festival" style tent not suitable for cold weather. It was bowed under rainwater and smelled stale. There were a number of black bin bags containing clothing.
The social worker later said: "I explained to her it was winter and the crowded space was wholly inappropriate for living."
The social worker also said Ms Marten told them she and Mr Gordon had an alternative lifestyle and asked them not to judge her.
The baby born in Wales in winter 2017, referred to in court as FF, was initially made the subject of an interim care order and Ms Marten lived with the baby in temporary mother and baby accommodation.
On two occasions social workers spoke to her about the risk of falling asleep with the baby on her chest.
It was explained that falling asleep with a baby on you poses significant risks due to the potential for suffocation, overheating and positional asphyxia. Their airways can easily become blocked when placed in certain positions, especially if they are sleeping on an adult's chest.
Also, the weight of an adult's body combined with the soft surface of a bed or couch can cause the baby's head to be positioned in a way that restricts their ability to breathe properly. This is known as positional asphyxia and can be extremely dangerous. Ms Marten expressed her commitment not to cause FF any harm, and promised to stick to the advice.
The jury was told that an order was made which allowed for FF to be cared for by the couple under social services supervision. This order was discharged in 2018.
After that the couple moved away from Wales, and had three more children.
But after the birth of their second child, referred to in court as GG, social services became involved again.
During one home visit, when Ms Marten was heavily pregnant, she hid her body behind a door and said she did not want social services to be "draconian".
At one point the couple left a child behind in hospital in order to attend a Family Court hearing, even though they had been offered a video link.
Once they had been separated from their children they continued to have "contact sessions" at which their interaction with the children was described as "excellent".
But their attendance at the contact sessions was inconsistent. The children were distressed by this. One child became quiet, withdrawn and inconsolable, saying on one occasion: "Mummy and daddy cancelled again."
After assessing the evidence a Family Court judge ordered that the four children should be adopted and care placement orders were made for all four children.
As well as manslaughter of baby Victoria, Ms Marten and Mr Gordon are also accused of four other offences: cruelty to their baby; concealment of the baby's birth; causing or allowing her death; and perverting the course of justice by concealing the body.
They deny all charges.
The trial continues.
