I don’t know if it has a name, but I’ll take that spicy mayo-ketchup-horseradish sauce that Outback serves with the Bloomin’ Onion and Burger King serves with their onion rings.
That was my first thought: what is it being paired with?
For things like meatloaf or shepherd’s pie, it has to be Heinz Chili Sauce, or HP Sauce, or Worcestershire sauce.
For pastas, it has to be a nice Bolognese or Marinara.
Soy sauce.
My version of Henry Bain sauce, it’s amazing on all kinds of savory dishes.
Something about the consistency makes Valentina stand out.
I like tomato ketchup with any dish including fries … and also shepherds pie.
But if the island offers unlimited spaghetti, I’d like carbonara sauce instead.
Mine! 
Lots of mentions of hot sauce but no salsa yet, unless my skimming of the answers so far was too cursory. So, a good tomato-based salsa.
Well, if I could only pick one, it’d be A-1 steak sauce for me. I’ve been known to pour and eat a spoonful even with no meat around. Love that stuff.
But I’d really miss Alfredo sauce on pasta, and teriyaki on couscous and rice, and BBQ on chicken…
If its an island, with the promise of fresh fish, then it has to be homemade fresh Kasundi - Wikipedia for me. MMMM MMM
Probably marinara, but a jug of Herdez green chili sauce would be good, too.
I choose to assume that this sauce is air-dropped daily via tiny parachute, mysteriously from the sky, warm in a thermos with a Dharma Initiative logo on it. That sauce is Bearnaise.
If I’m trapped on a desert island, I assume I’m probably going to be surviving like Tom Hanks in Castaway, eating mostly seafood and coconuts. So I’m thinking in this specific scenario, I would want cocktail sauce to go with my crab.
Now if a pallet of chicken wings that magically stayed frozen until I was ready to cook them washed up on shore, my answer would be different.
Is just me alone on the deserted island or are their others?
If it is just me probably ketchup.
If there are others what’s the best sauce to go with long pork?
If this is the case, then The Macallan is a sauce!
Which brings us back to A-1… 
That’s the great thing about Valentina. It goes great with all of them—seafood and chicken wings alike!
Either Daddies or HP. Probably HP.
Hot fudge. Because I will need a huge reward after eating fish. I hate fish.
Briny_Deep’s answer is the correct one. Bearnaise sauce is heavenly and I could eat it on anything. If the island came furnished with filet mignon steaks, that’d be a definite plus.