By JULIA MOSKIN

It’s not easy being a Mexican restaurant in the United States. Some customers are outraged if the menu doesn’t have burritos; others, if it does. Some see authenticity when chips and salsa aren’t on the table; others see a rip-off.

“Everybody here thinks they know what Mexican food is,” said Nicholas Cox, the chef at La Esquina in downtown Manhattan. “Especially if they’re from Texas or California.”

Salsa, in particular, has emerged as Mexico’s most misunderstood culinary export. In Paris, Mexican restaurants make it with minced cornichon pickles and ketchup; in Japan, with green shishito peppers and Kewpie mayonnaise; in American factories, with corn syrup and red bell peppers.

Soon after the United States “discovered” salsa in the 1980s, it soared to popularity, famously outselling ketchup by 1992. American cooks flirted with peach salsa and corn salsa, while supermarket salsa evolved into a thick, sweet mix.

But on its global journey, salsa as it is actually made in Mexico often became lost.

Irma Verdejo, an owner of Tulcingo del Valle in Manhattan, said that her customers often see it as a generic mixand- match condiment. “I fight with people about salsa all the time,” she said. “They want to put this salsa with that dish, or they want it more spicy, or less spicy. And they always think it should be free.”

Salsa is a cornerstone of the Mexican kitchen, a contrast for a repetitive diet of corn, squash and beans. Just as French cooks understand that bearnaise sauce suits some dishes, hollandaise others, Mexican cooks know salsas have different qualities and functions.

Recently, true salsas have been finding their place at the Mexican table in New York. “Now there is a conversation between Mexico and the United States about what is good Mexican food,” said Iris Avelar, who grew up moving between Guadalajara and the San Francisco Bay Area, where her parents had a restaurant. She is an owner of La Superior in Williamsburg, Brooklyn, which specializes in carefully made Mexican street food.

“When my parents opened their restaurant 20 years ago, we had to have a ‘gringo salsa’ with no chilies at all,” Ms. Avelar said. “That would never happen now.”

Flavorful salsas, made in the classic way, have become much easier to find in New York, both at hipster magnets - like La Superior; Cascabel Taqueria on the Upper East Side; Mercadito in the East Village; Barrio Chino on the Lower East Side; and Hecho en Dumbo on the Bowery - as well as at traditional places like Tulcingo del Valle, as well as Tortilleria Nixtamal and Tia Julia, both in Queens.

The chefs in these kitchens create salsas that go way past “red or green” and “mild or hot.” They highlight the green, herbaceous flavors of fresh chilies and the raisiny sweetness of dried; they use heat judiciously, to contrast the richness of meats like house-made chorizo or braised veal tongue .

“A great taco is a perfect food,” said Danny Mena, the chef at Hecho en Dumbo, who is from Mexico City. “A tortilla, a protein and then the salsa is just the genius Mexican way of seasoning every bite with acidity, heat and salt.”

The word “salsa” does simply mean “sauce” in Spanish, but in truth salsa in Mexico almost always means there will be chilies. It is meant to have flavor and depth, with a mix of tart, sweet, salty and hot effects. It should be “poignant” with heat, Mr. Mena said, but not hit you over the head with it.

Most authentic salsas are based on just a few ingredients - chilies, tomatoes, onions, garlic . What produces the dizzying range of flavors, tastes and textures is how those ingredients are cooked (or not) - charred on a dry surface for smokiness, shallow-fried in oil for creaminess, simmered in water for brightness or simply buzzed in a blender for punch.

In modern Mexico, salsa often gets extra savor from salsa inglesa (Worcestershire sauce), salsa de soya (soy sauce) or jugo Maggi (an MSG-based sauce).

“Salsa is not even vegetarian in Mexico,” Mr. Mena said in discussing what makes salsa genuine. “How can anyone know what authentic is?”