The stocktake, which launched in April 2017 and was carried out by longstanding government adviser Sir Martin Narey and children's social worker Mark Owers, makes a total of 36 recommendations.
Alongside suggesting a national register, it also calls for foster carers to receive better support, and for improvements to be made to foster placement commissioning.
The report states that although more carers are needed, there is not an "absolute shortage".
"The overwhelming majority of children needing a fostering placement on any one day are placed," it states.
"Indeed, at any one time, there are about 16,000 fostering households without a child living with them. The shortages are down to geography or the availability of carers who can look after more challenging children. This means that, too often, matches are made between carers and children that are not ideal and, after a short period, the child has to be moved again."
The report states that there is "great merit" in developing a national register of foster carers potentially holding details such as their address, the date of their approval as carers; the number of beds and bedrooms in their home; the number of vacancies for children; their personal characteristics such as age, gender, ethnicity, religion and language; and their level of training and expertise.
"A number of organisations [have] called for a national register of carers," the report states.
"Such a register would provide vital information which could improve recruitment. And such a register could also provide a vacancy management system and radically improve matching."
The stocktake also called for improvements in commissioning foster care placements by local authorities - stating that current practice is "gravely limited" by the fact that 152 individual local authorities commission from independent fostering agencies (IFAs).
Local authorities in England place about 66 per cent of children in placements they manage directly and place the remaining third in placements provided by IFAs which mainly operate in the private sector.
But the report claims there is a "startling failure" to obtain best value from the market, adding that the providers, not the purchasers, "too often have the upper hand".
"We saw virtually no evidence of discount pricing for large numbers of placements from the same provider and framework contracts, set up after long and tedious processes," the report states.
"These were routinely ignored in favour of more expensive spot purchase arrangements. Local authorities need to come together into about 10 consortia and negotiate with IFAs to provide placements at significantly reduced cost, almost certainly through guaranteeing them a certain level of business. The routine absence of such arrangements is extraordinary."
The report adds that although only three local authorities are close to being self-sufficient on foster carer provision (recruiting more than 95 per cent of the carers they use), many more - if they were sufficiently determined - could join that number and better exploit the back office economies of scale.
It adds that there is "considerable scope" for local authorities to follow the example of Peterborough, which has agreed a long-term deal with The Adolescent and Children's Trust to run its fostering and adoption services.
"[Local authorities] should not be discouraged from making such partnerships with IFAs from the private sector whose quality of care, and an ability to find homes for the most challenging children, is not in dispute.
And the report also suggests that local authorities should be allowed to ditch the independent reviewing officer (IRO) role.
"Our conclusion is that, despite the commendable commitment of some individuals, there is little to recommend the IRO role and believe local authorities should be allowed to dispense with the role; reinvesting savings in frontline staffing," the report states.
Sir Martin Narey said: "Foster carers make a remarkable contribution to the lives of children often damaged by neglect and that contribution needs to be recognised.
"But fostering can be made even more effective, and could make an even greater contribution to the welfare to some of the country's most disadvantaged children.
"Foster carers must be allowed much greater authority in making decisions about the children in their care and they need to be liberated to offer the physical affection which is a vital and necessary part of most children's healthy upbringing.
"We make 36 recommendations and if all were to be implemented, as I hope they will be, then local authorities will have foster carers who are better motivated and better appreciated."
Children's minister Nadhim Zahawi said: "The report gives us an opportunity to celebrate foster care and to recognise the invaluable role foster parents play in the lives of vulnerable children.
"We are committed to supporting them in this role, and that's why we recently announced that we will extend our 30-hour childcare offer to foster children to provide extra help for foster parents.
"We will carefully consider the review's recommendations, alongside those from the education select committee, over the coming months to determine how they can help us to make sustainable improvements to the fostering system and to the outcomes for looked-after children."