Illustration by John Dykes

We are so CLOSE I can, regretfully, smell it. My wife and I are in the back of a Manhattan bodega — the particular kind of corner store found everywhere in New York City — where we discover a tantalizing scene.

Nestled beyond the lottery tickets, beers, batteries, Little Debbie snacks and a fruit display consisting of a few limes and one sad banana, there it is: a used cat litter box and a half-eaten bowl of food.

But no bodega cat to be seen.

There’s a great equalizer in New York City. Even Gotham’s billionaires sometimes need orange juice and a pint of Ben & Jerry’s at 1:30 a.m. Here, instead of driving to the 7-Eleven, you stagger to the bodega, the ubiquitous, largely independent, often-24-hour everything store.

How do you know it’s a bodega? If the bottom of the store’s awning displays a desperate attempt to list everything it stocks — cold beer, soda, food stamps, cigarettes, candy, ice cream, juice in tiny plastic barrels, toothpicks, human teeth, the collected works of Allen Ginsberg — then it’s probably a bodega.

Ditto if its outside windows are plastered with beautiful, enticing photographs of food items that can’t actually be purchased inside the store.

And last but hardly least: if the store has a cat.

Note to New York City health inspectors en route to Aruba to get away from it all: Close this magazine now.

Other readers: Yes, although it is technically illegal, bodegas, like pirate ships, are typically stocked with cats to control rodent populations. When you’re not actively looking for a bodega cat, they are omnipresent, most often sprawled on a stack of newspapers so you have to do a magician’s tablecloth trick if you want to see the score of yesterday’s game.

But go looking for them, and the Global Cat Cabal apparently files an emergency injunction to make all of its bodega constituents suddenly disappear.

My wife and I have a plan: See how long it takes to spot one bodega cat in each of the city’s five boroughs. Our lone rule bars us from asking the store owners if they have a cat because they might think we work for the city and would try to bribe us … or they might whack us with a broom.

We start at 2 p.m. in Manhattan. Our close call with the foul-smelling cat litter box comes several bodegas in.

Frustrated, we head to Washington Heights — one of Manhattan’s northernmost neighborhoods and the bodega-cat seat of power. Old men are playing dominoes outside the first bodega we see; inside, patrons are drinking beer wrapped in paper bags.

We buy a bag of Utz Bar-B-Q potato chips and a blueberry cream Soda Shaq. As I’m paying, Jenny taps me on the shoulder. An angelic light shines down on a cat snoozing discreetly on a shelf.

His name is Mizuri.

Next stop: Da Bronx. Perfectly, a joint called Dolce Vita is piping Frank Sinatra onto the street. We spend hours wandering into bodegas, munching a giant eggplant Parmesan sub and downing espressos in Styrofoam.

Then I hear it. From beyond the Ramen noodles, a meow. By the time I scramble to it, there is only a swinging door. But we’re going by bird-watching rules, which say a species is considered spotted if you simply identify its call. So we consider the Bronx conquered.

We have an inside track on Queens, New York’s most diverse borough. A few months earlier, I photographed the incredible sight of two cats lounging in one bodega. Now we use my ­Instagram as a road map to the bodega-cat equivalent of a double rainbow, crossing off Queens while purchasing black-bean sauce and rice noodles.

Whenever I tie up the leash for Murray, my dog, outside my neighborhood bodega in Brooklyn, its guard cat sidles to the door, raises its back and glares at him for the duration of my shopping. So now we stop at home, pick up Murray, rush to the bodega, start to tie up his leash, see the angry cat’s ears as it steps to the doorway and rush back home to redeposit Murray in record time.

All I know about Staten Island I learned from Wu-Tang Clan rap albums, which make little discernible mention of bodega cats. So we’re flying blind as we drive over the Verrazano-Narrows Bridge.

After a late dinner made by real-life Italian grandmas — look up the restaurant Enoteca ­Maria — we head into a bodega named Not Guilty Inc. for a dessert of a black-and-white cookie. No cat, though.

It’s 12:30 a.m.— more than 10 hours after our quest began (try to beat our time if you dare) — when we step into the next bodega and are suddenly greeted by a fat, purring tabby standing in front of the door like he’s been expecting us for eons.

For legal reasons, we’re not identifying any of the bodegas housing illegal feline squatters. But I will disclose that the name of the Staten Island bodega included, appropriately enough, the word lucky in its name.