Call it the pied-à-terre surprise: Gov. Kathy Hochul’s office is still figuring out how her gift to Mayor Zohran Mamdani will actually work.

That’s one problem with a levy made up to fit a headline (“make the rich pay”) and a revenue figure ($500 million a year), with zero regard for making any sense.

Most of Hochul’s multibillion-dollar bribe to Mamdani was simply permission to borrow unwisely by deferring pension-fund contributions, plus some direct state spending; the pied-à-terre tax was mostly about the symbolism.

That is, it let him make that nasty video outside Ken Griffin’s home, so the mayor could assure his core voters he was hitting fat cats even if Hochul wouldn’t buy his dreams of income- and corporate-tax hikes. 

Nobody thought about complications, such as how it could leave co-op boards on the hook while they try to recover the surcharge from wealthy absentee apartment-owners.

Basically, if the surcharge applies to coop units serving as out-of-towner crashpads worth $5 million or more — the gov’s office doesn’t seem certain about that — then non-wealthy, fully NYC-based coop members could get hit with stiff temporary assessments while the board dings the “bad rich people” for their payment.

Co-ops don’t have separate tax lots for individual apartments: The whole building gets assessed as a single property; the co-op pays the real-estate taxes and apportions the charge per share via monthly maintenance charges.

The co-op can’t delay paying, so it may have to make others cover the surtax while it gets the absentee owners to cough up their share; in buildings with just a few units, the added charge could devastate the unlucky non-rich.

The gov’s office told The Post the law “includes tools for the city to directly enforce against the unit owner,” but doesn’t seem clear on how that applies to people who don’t own their units (but only co-op shares).

Hochul’s minions also offered lots of definitive-sounding statements about how “nothing in this policy would diminish housing options made available to New Yorkers” and boards won’t be “penalized,” but couldn’t seem to explain exactly how the scheme would work.

These concerns have some boards are already looking at banning future pied-à-terre ownership to avoid the risks, no matter how much that might depress the building’s value.

Mamdani’s people don’t seem worried; after all, the mayor’s already scored his social-media win — what else really matters?

Between the mayor’s cavalier attitude about paying for his agenda and the gov’s willingness to sign off on schemes that no one’s truly thought out, the city’s in for a lot more nasty surprises.