I keep tartar sauce as long as I would mayo, but if I'm making aioli, I tend to make the mayo myself and use olive oil. If you are making it with commercial mayo, there should be no reason to make a larger quantity than you need at the time, but I don't think the garlic will make it go bad faster than the mayo, although the flavor will diminish over time.

If commercial mayo is used as the basis for the sauce, it should last as long as the mayo.

Commercial mayo, depending on the brand, may have stabilizers. They keep it from separating. It's also made with pasteurized eggs.

Scratch tartar sauce might separate. If you have a high speed/commercial blender, it may emulsify well enough that it'll last longer than you'd want without separating. That's suspending tiny bits of water in the fat. It's a lot harder to do to that degree with just a whisk.

The bigger issue is what bacteria and mold you're introducing just through adding ingredients, using tools, exposing it to kitchen, etc. That's going to be the limitation on the storage life. If it smells funny or looks funny or looks furry, throw it out, same as always. :) But tartar sauce isn't a big one for spoilage as has been said above. You can pasteurize your own eggs sous vide or by steaming.

Re aioli, are you talking about flavored mayonnaise? Real aioli, which is an emulsion of just garlic and oil, should not be kept more than four days in the fridge and never at room temperature. It can grow botulism. I would think garlic in mayonnaise would have the same issue. Roasting the garlic doesn't help. Other than garlic, however, you can store flavor mayonnaise as long as the stuff you put in it would last, at least.

In my inquireys regarding stuff made with commercial mayo and safety, I found that not only the stablizers and preservatives but also the acidity of mayonnaise prevent bacteria from growing. The word is that salads with mayonnaise are not a source of food poisoning as previously thought.
As for how long the tartar sauce will keep in the refrig, that depends on what you add to it. Onions might be suspect unless you also add more lemon juice....but pickles are fine.
Acid inhibits the growth of botulism spoors....which is why pickled beets are OK but just canned beets must be pressure canned to be safe.

IMPE, fresh garlic in dressings and cold sauces develops an 'off' flavor over time. It may not be harmful to eat, but it doesn't taste as good as freshly made.

I know....took me a long time....but what is a few days?
Working in the church kitchen I see people paranoid about getting the potato salad into the refrig, but very unconcerned about the chicken!!
And mayonnaise' keeping qualities is another reason to eschew Miracal Whip.

I make remoulade with commercial mayo. What we don't use the day we boil shrimp keeps for about a week in the fridge. By then it's all gone. ;)

What chase said. And not just for mayo. A week would be a stretch for anything “made” to live in my fridge.

Mayo as the main culprit for food poisoning is out of date. Plain cooked potatoes, rice, even sliced tomatoes (they've bred too much of the pathogen-preventing acid out of them) are all potentially dangerous if left at room temp too long.

Re: mayo based sauces, if I'd added fresh herbs I'd keep it several days and then let it go.

Pillog...interesting about aïoli ...in Provence I've seen it made with bread soaked in water, squeezed dry, garlic added then the oil. All done in a pestle and mortar.

Garlic added to any mayo......no. I wouldn't keep it.

Islay, the length of time it's okay to keep garlic in oil in the fridge comes from the University of California, Davis, where they have the agriculture school. That's a reliable source for food safety.

Re the method for aioli, I've read that French versions are different and stretch the definition. Garlic and oil only is Iberian, I think. Around here, it seems like chefs call any oil based emulsion aioli, to the point where it's meaningless. I'm used to it now, but when it first started, I was thinking (I hope not aloud!) WTF? The French version you described with the bread still sounds close enough to be called aioli (literally garlic oil). Okay, I looked it up, and am informed that aioli is Catalan and Provencal. I have no information about whether with or without bread came first (nor do I care!), but the addition of egg or mustard to aid the emulsification is definitely new. :) And as far as I'm concerned, might as well be mayonnaise. :)

I rarely buy commercial mayo but do date the top when opened. Said to be good for 4 months in the fridge.

Storing Garlic in Oil

Take extreme care when preparing flavored oils with fresh garlic or when storing fresh garlic in oil. Peeled garlic cloves may be submerged in oil and stored in the freezer for several months or in the refrigerator for no more than 4 days. Label refrigerated garlic in oil mixtures to show the preparation date (or preferably with

a “discard after date” label). Do not store garlic in oil at room temperature. Garlic in oil mixtures stored at room temperature provide an ideal environment for Clostridium botulinum to grow and produce toxin (low acidity, no free oxygen in the oil, and warm temperature). The same hazard exists for roasted garlic stored in oil. At least four outbreaks of botulism associated with garlic in oil mixtures have been reported in North America in the late 1980s and 1990s. Outbreaks in 1991 (California) and 1999 (Florida) were associated with garlic in oil prepared in the home.

linky, HERE

I have a personal limit for anything homemade...sauces using fresh ingredients like yogurt, mayo, miso, dressings,...3-4 days as they tend to loose initial freshness.

Exceptions are heavily vinegar/citrus based dressing seem just as good for 5-7 days.

Dried and otherwise preserved garlic never tastes as good to me as fresh. I very rarely use anything else.

Rachael Ray (I know, I know....not health food!) uses a vegan mayo in things like potato salad to be consumed at an outdoor picnic. I haven't tried it, but it sounds like a good idea for safety.