Tomato pasta sauce splits. Why?

If by "fresh tomato paste" you mean you're starting with raw tomatoes, that's probably your issue. Most tomatoes you grow in your garden or buy from the store are salad tomatoes - they're not meant for cooking, and will result in a watery sauce like you describe.

You want a "sauce" tomato. Roma tomatoes or almost any plum-shaped tomato work well. There are others, but they're harder to find.

A basic puree of nothing but fresh tomatoes is a simple suspension -- solid particles floating in a liquid substrate. Over time those solids will settle out, leaving a clumpy solid layer and a watery liquid layer. Adding more water (e.g., from pasta water or incompletely wilted onions) will accelerate the trend. To avoid this result, you have a few options:

1. Remove the water upfront. Before you start cooking, place a large strainer over a large container, line the strainer with a double layer of cheesecloth, and fasten the cheesecloth to the strainer with clips. Pour the tomato puree into the strainer and refrigerate 20-30 minutes, stirring once or twice. If the strainer isn't big enough to hold all of your puree, do it in batches. Alternatively, you can simmer the sauce for a while, until it reaches your desired level of thickness.
2. Create an emulsion. Essentially, you are binding up the water phase with fat using something like egg yolks or whipping cream. Check out recipes for "alla vodka" sauce or "tomato alfredo" for examples.
3. Use a starch thickener. This is not my favorite, but again, starch binds up water. There are many ways to do this, such as using bread or tortillas (see: romesco sauce), but you might also see it done in alfredo-like sauces using, say, a starch-based paste like a roux. The theory behind adding pasta water is that the bloomed starch from the pasta itself will help bind up the sauce. The thing is, if you boil your pasta in plenty of water (to avoid boil-over), there won't be enough starch in the water to do the job. Moreover, that starch has already fully hydrated, so its ability to thicken a sauce will be very limited at best.
4. Reduce the proportion of water to solids. Tomato paste will absorb a little of the water released by the puree. This strategy will only work at the margin, however, unless you add enough paste to overwhelm the flavor of the fresh puree.

(Edit: I'm assuming you are using only a tiny amount of olive oil to sweat your garlic, such that you are not talking about having an oil slick over your sauce. If that's the problem, the answer is trivially simple: use less oil!)