Think Australians can never agree on anything? A whopping 81% of Australians support the government’s proposal to take away the citizenship of dual nationals who are “engaged in terrorism or supporting terror groups”, today’s Essential Report finds. And 73% of Australians support stripping citizenship from sole nationals if they are eligible to become citizens of another country.

The strong result vindicates Tony Abbott’s decision to try to use the issue as a wedge against both his internal critics and Labor, which backs the idea in principle but wants to see the legislation. Liberal/National voters are the voter segment most overwhelmingly in favour of stripping citizenship, with 93% supporting it for dual nationals and 90% for sole nationals. Greens voters are the most opposed, though even 52% support stripping citizenship from dual nationals engaged in terrorism, against 35% who disapprove. And almost half (44%) of Greens voters support stripping citizenship for sole nationals, against 31% who disapprove. Labor voters are right in the middle, though they also strongly support stripping citizenship from both dual nationals (77% for, 11% against) and sole nationals (67% approve, 15% disapprove).

Q. Do you approve or disapprove of the Government’s proposal to take away the citizenship of dual nationals who are engaged in terrorism or supporting terror groups?

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Q. And would you approve or disapprove of taking away the citizenship of sole nationals who are engaged in terrorism or supporting terror groups if they are eligible to become a citizen of another country?

But even though there is strong support for the measure, most Australians (including most Liberal voters) think the power to strip citizenship shouldn’t rest with the Immigration Minister — an issue that split cabinet. More than half (54%) of those surveyed think the decision to strip citizenship should be made by a court of law, not a government minister, compared to 24% who want the decision to be made by a government minister, and 22% don’t know). This is true across all voting segments, though Liberal/Nationals voters are most likely to want the decision to be made by a government minister (34%, against 47% who want the decision to be made by a court), with Greens voters most in favour of the courts deciding a citizen’s fate (71%, against 8% who think the decision should rest with a government minister). Labor voters also strongly favour a court making the determination (58%) over a minister (22%).

Tough talk on terrorism may be helping the Prime Minister, with Tony Abbott continuing his recovery in the eyes of voters: Abbott’s approval rating is up three points to 39% since May. Half of Australians still disapprove of his performance as Prime Minister, but that is down four points from 54% in May. Abbott’s net approval rating has risen from -18 to -11, which is his best net rating since November last year. And the news gets worse for Bill Shorten, whose approval rating held steady at 32% but whose disapproval numbers are up 4%, to 45%. His net approval rating has dropped from -9 to -13, his worst since becoming Opposition Leader.

Essential also asked about rising house prices. Just 23% of voters think rising housing costs are good for the economy, versus 36% who think it’s bad for the economy. Rising prices are recognised as good for home owners (49%) and investors (46%), but 74% agree they are bad for first home buyers and 57% say they’re bad for average Australians.

Voting intention has remained steady again: the Coalition primary vote remains on 41% and Labor remains on 40%. The Greens are down a point to 9% but the two-party preferred result remains the same on 52%-48% in Labor’s favour.