He hovers at my side, eyeing up my picnic with lusty envy. In front of me is a mound of glistening olives, plump sun-dried tomatoes, bread and cheese: my lunch. Or is it breakfast? I've lost track already. It's 11am in Taverna Azzurra, a local bar where soft rock blares from the speakers, and wine pours from repurposed water coolers. Behind me a greying, rotund man lunges in to nab a lump of cheese. Stealthy? No. Sozzled? Oh yes.

Anywhere else in the world, having your food pinched by a drunk loiterer would be a sign to get the hell out. In the Sicilian capital, Palermo, it just slots into the city's gritty, convivial street food scene – and is hardly surprising in a place where the local saying is 'bonu vinu fa bonu sangu': 'good wine makes good blood.' And bad habits, perhaps.

If eating out in Italy conjures up images of lingering al fresco pasta marathons or afternoons snoozing off pizza, forget it – Sicily's port city does things its own way. Table and chairs? Grow up. Knife and fork? It's fingers only for you, my friend. Here in the island's capital, food is devoured standing up and on the streets, and according to one of the great Palermo proverbs, 'a person eating must make crumbs'. Clearly, I'll fit right in.

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What will become something of a foodie epic had started two hours earlier, at 9am in the morning. Accompanied by my boyfriend, Adam, I meet Marco Romeo – the owner of Streat Palermo food tours and the Godfather (excuse the label, but it is apt) of Palermo's pavement stalls. I'm hankering after the silky hit of an espresso, a habit I've become accustomed to on Italian breaks. "Never!" Marco stops in his tracks. "Mix it with all this food and you'll have a dodgy stomach."

That's the least of my worries. I smell it before I see it – the pungent waft of boiled meat. A man called Mr Toni greets us with a nod before submerging his grubby hands beneath the floral cloth 'lid' of his wicker basket. He rummages around before pulling out soft, slippery folds of squidgy yellow... something. "Leftover veal!" Marco smiles triumphantly, sprinkling our paper cone of frittola – that's fat and cartilage infused with bay leaf – with lemon juice. "Boiled, then fried in lard, then boiled again. This is typical Palermitan street food." Dear God.

Forget cheerios, veal is the standard breakfast here in Palermo

Forget Cheerios – this is the standard breakfast in Il Capo, the 1,200-year-old main fish market in central Palermo, a place marked by the culinary footsteps of former Sicilian conquerors including the Arabs, Spanish and Normans.

Take pasta chi sardi – the city's long-standing signature dish. It's a prime example of how Palermo's food is a mash-up of historical influences – sardines bought from the market are tossed with wild fennel from the Sicilian mountains, along with saffron, pine nuts and sultanas (traditional flavours from the Middle Eastern conquerors), and of course, spaghetti.

Arabian influence is everywhere. It's infused in the local dialect, the design of the alleys and the ingredients at our next pit-stop, a hole in the wall called Da Arianna, the home of the arancina.

Brought to Sicily by the Arabs in the ninth century, the minced veal and saffron arancina (a regional tomato-free version, hence the spelling with an 'a') was originally served like risotto, before locals realised a rounded shape would make it an easier snack to carry whilst hunting. Today, the family who run Da Arianna churn out hundreds of the orange-shaped balls a day. Mamma expertly caresses grains of rice in the back kitchen, while Jesus looks on from a pink, plastic cross – typical of a hugely Catholic city boasting 220 churches.

Arancina, deep-fried minced veal and saffron deep-fried risotto balls, are an essential street food in Palermo

The most beautiful but rarely visited of these is the baroque Immacolata Concezione. While our next dish bubbles away, we nip inside to see the gleaming statue of Saint Rosalia, the patron saint of Palermo. It's a temporary break.

"All this food is fried," Marco informs us as our next dish arrives. He's nervous we might dismiss it as unhealthy. Deep-fried? Pah! It's our favourite vice! Adam and I bat his worries away, before burning our mouths on fresh-out-the-fryer morsels: salty, crisp chickpea fritters known as panelle, and stubby, phallic potato croquettes called cazzilli – deriving from the Italian for penis, of course…

Following Marco, we negotiate the labyrinth of Palermo's backstreets, where signs of destruction are still as evident as if the 1943 allied invasion had happened yesterday. Walls crumble around us, and rotting beams support precarious buildings, where you hurry past in case they collapse (which they frequently do, Marco reveals).

Out in the open, danger comes from speeding mopeds, so we seek sanctuary on the kerb with Mr Mario (so well known he doesn't need a surname), who is famed for his sfincione cart.

The word is derived from the Latin spongiam, or sponge. And that's what it is, sort of. A hunk of pillowy bread, it's soaked in lashings of oil, tomato and pepper. Surprisingly delicate, it turns out to be the most unlikely Italian pizza I've ever tasted; in one epic bite it sticks two fingers up at the pink-hued-piazza and smooching-couples stereotypes of most Italian cities.

Market stall

The market overflows with a rainbow of fresh produce

We amble to Vucciria, where deteriorating houses are broken up by occasional streaks of colour. Piazza Garrafello reveals a typical Palermo scene – a building that's remained largely untouched since the 1943 bombings, held upright by rows of scaffolding.

Painted on the crumbling walls behind is some floral-inspired graffiti, a tribute from the Austrian artist Uwe Jäntsch who fell in love with the piazza when he moved to the city. The locals couldn't care less; in one corner of the empty square, two men slump against plastic chairs, peeling potatoes with a backdrop of long-abandoned balconies and half-arsed graffiti scrawls of PREGO PREGO PREGO, the ubiquitous Italian saying.

Mr Giuseppe (I'm not making up names) is our next port of call. He's responsible for the city's most famous fast food dish, pani ca meusa: spleen sandwich.

Food stops

Osteria Lo Bianco

Join the Palermitans at this bustling osteria for some of the best pasta and fish in the city. Book ahead.

Altri Tempi

Tuck into huge bowls of the local speciality pasta chi sardi while the Italian version of X-Factor blares in the corner.

Savoca Rosticceria

Stop at this locals' favourite bakery for a selection of high-end Sicilian bites including mini-calzone.

Rosciglione

The best place in Sicily to get cannolli with ricotta cream. The factory makes thousands of the sweet pastries every month.

A staple for Palermo's Jewish community in the early 1900s, the veal spleen, lung and throat cartilage is boiled and fried, stuffed into a bun that's been gently soaked in fat, then topped with ricotta cheese. "It's just like a burger," Marco nods, tucking in. Except, it's really not. Still, it's definitely got something. After recovering from the initial apprehensive bite, I happily scoff the rest.

On every corner you see that street food is a source of local pride – granddads stand at buckets mixing batter for hours, or fan kerbside barbeques. Their passion, it seems, is only rivalled by one other thing.

"Welcome to Palermo's second most important cathedral," Marco announces, waving towards an eclectic jumble of architecture. The first – Stadio Renzo Barbera – is down the road, still ringing from the din of the previous week's 3-1 win over Napoli.

But it's nothing compared to the real cathedral. Its dazzling facade sports Moorish and Baroque flourishes, dust-coloured Gothic-Catalan arcades and 12th-century Norman towers spiking into the sky. Inside is surprisingly neoclassical and understated; the biggest embellishment is the tomb and a shiny, gargantuan poster of Giuseppe Pino Puglisi, the priest who openly criticised the Mafia, and paid for it with his life in 1993.

Nowadays, however, the only danger for tourists is high cholesterol. "I'm adding you to my spreadsheet of tourists I like," Marco tells us, and suggests we meet later, after a pause from the onslaught of food.

A street-food trader selling bread

We set about exploring the city's Greek columns, Roman arches and Norman churches, but Sicilian food proves strangely addictive. We only manage five minutes before we're distracted by the sherbet-hued ice creams in a side-street gelateria called Gelato. Pistachio ice cream bulges from the side of our brioche buns, a challenge to eat while negotiating cramped alleys.

Massive knickers dangle from washing lines, abandoned mattresses lean defeated against walls and bin bags erupt like nearby Mount Etna. One lane is blocked by a clapped-out Fiat 500 – a photographer's dream. Palermo's gruff streets won't be winning any beauty pageants, but the dishevelled vibe is no less alluring.

It proves even more so in the market area of La Vucciria, which is transformed by the time we return that night. Makeshift barbecues line the perimeter of the small square, and billows of smoke wafting from the coals mingle with that from the cigarettes and spliffs of the city's youth, who seem all too ready to abandon mamma's home cooking.

Remarkably, we've still got some stomach space to spare. But once again the dish in my hands is unrecognisable – curly fried knots of something grey and bobbly. "Veal intestine!" Marco cries for the third time that day. I'm unfazed now. With any reservation dulled by numerous bottles of cheap Forst beer, the smoke-infused meat squirted with lemon tastes good. Really good. I could get used to this.

Ice cream bulges out the side of brioche buns

Grabbing a helping of fried bacon wrapped around shallots for the road, Marco leads us back to Piazza Garrafello. Although deserted earlier that day, it's now packed with tables, pop-up beer carts, decks and flashing lights, the scaffolding threatening to topple with every beat and footstep. The surrounding cobbles throng with lively beer-chugging locals and men enthusiastically playing table football underneath a centuries-old statue of the Genius, the emblem of a city whose Latin inscription warns: 'Palermo devours its children, and feeds its foreigners.'

Hotel Principe Di Villafranca

Away from the crumbling streets of the old city, but still brilliantly located for central access, this cosy boutique hotel is the perfect spot to spread out after your food fest. Around the corner you'll find Palermo's best bakery: Savoca is a local favourite, where counters are lined with doughy, tomatoey and cheesy treats. You can even bring a takeaway box back to your room. But who in their right mind would do that?! Erm... Three nights with flights from £572pp. classic-collection.co.uk

No surprise, then, that the evening climaxes, as always, with some late-night indulgence. Standing at the counter of Fratelli Ganci, we share doughy snack-size calzones and arancina before a final spiedino – a deep-fried (of course) brick – sorry, pastry – of minced meat. It's so substantial and dense that the paper plate strains from the weight of it. As, unfortunately, do my jeans.

Cue an emergency digestif. We squeeze back into our earlier hangout, Taverna Azzurra, which is now packed with a nimble, younger crowd in Converse high tops. After 10pm, the bar owner's sons run the show. My food-nabbing mate and his paunchy pals are long gone following a hard day guzzling sweet, locally-produced Marsala, Zibibbo and Sangue de Sicilia wine. It would be rude not to follow suit, wouldn't it? We order a round. Hey, it's good for the blood. ■

Food tours are available with Streaty, streaty.com; arrange flights and accommodation with classic-collection.co.uk.