They also see eye to eye on having a much more rigorous national interest test for foreign purchases of farmland, an approach not entirely without support even within Federated Farmers, among those who worry about the future of the family farm.
However, banning non-residents' purchase of housing would require the renegotiation of New Zealand's free trade agreement (FTA) with South Korea.
Under the earlier FTA with China, any future New Zealand Government could ban the sale of houses or farmland to Chinese investors, and China has reciprocal rights.
But the China FTA also has a most favoured nation clause which requires New Zealand to treat China no less favourably than it does other countries in any subsequent agreements. Both the Korean FTA and the Trans Pacific Partnership (TPP) agreement do that, in effect ratcheting up Chinese rights too.
Trade Minister Todd McClay dismisses as fanciful the idea that the Koreans would be interested in reopening the agreement so soon.
However the TPP agreement, which is still being amended in hopes of salvaging it after the United States pulled out, does have a carve out of sorts attached to the investment chapter: "New Zealand reserves the right to adopt or maintain any taxation measure with respect to the sale, purchase or transfer of residential property." Australia's more populous states have such a tax on foreign investment in residential property and in principle it could be set at a seriously discouraging level.
The fact that the carveout was incorporated in the TPP text by a National Government suggests that that party is not fundamentally opposed to the idea.
So we might see something there.
Another longstanding New Zealand First policy, dropping the GST on food, has been quietly scaled back. In the past it costed the policy at $3 billion a year (to be funded by cracking down on tax evasion), which would be consistent with the only exceptions being restaurant and takeaway meals.
But in a Peters speech last month, that had become dropping the GST on "basic, essential foods" - which are they? Ask your grandmother! - and the price tag cut to $600 million to $700m. Shrinking the scope of the policy, and its revenue cost, by four-fifths suggests it might be one he is serious about.
New Zealand First also wants to drop the GST on local body rates - a tax on a tax - which would cost around $500m a year. It is a policy which might appeal to the older members of its constituency.
A consistent theme in Peters' speeches in provincial centres has been a policy of returning (somehow) to the regions the GST paid by tourists visiting those regions, to fund roads and other infrastructure. An Auckland Council bumping up against borrowing limits might welcome that as well.
He put the cost of that at $1.5b and rising.
So this is mounting up - $2.7b out of a GST take of $20b a year.
Jacinda Ardern has made it clear that any agreement with Labour and the Greens would have to conform to the budget responsibility rules those two parties have agreed, which includes a net debt target of 20 per cent of GDP (in five years' time).
Likewise, it is hard to see Bill English suddenly turning fiscally profligate.
Both of them should remind Winston Peters that most of the government debt is held offshore so any measures which increase it, all else being equal, would increase the future flow of tax dollars in interest payments to foreign bondholders.
On immigration, New Zealand First wants caps on family reunification visas and on the older immigrants it sees as a burden on the health system.
It favours raising the minimum residency requirement for eligibility for New Zealand Superannuation from 10 to 25 years (after the age of 20). That is also recommended by the Retirement Commissioner.
Some policies can probably be disregarded.
The tertiary education policy of making student allowances universal, paying 100 per cent of fees and forgiving student debt if they remain in the country for the same number of years they were students (or forgiving every other dollar if they are already paying off the loans) would be pretty expensive.
And Peters has walked back the policy of reversing the privatisation of ECNZ on confiscatory terms, to one of buying back the shares "when the market price is right".
The foregoing is a random selection of policies from a long list.
How many of them see the light of day remains, of course, to be seen.