Recently Xiao Hei, my black VW Jetta of a certain age, has developed a fuel flow problem. When the tank is less than half full, the fuel supply becomes erratic, and the engine starts to stall, and then die. It always starts up again, but then runs sputteringly. I wonder if it’s possible that there is dirt/sediment/rust flakes in the fuel tank that are clogging up the intake. The less gas there is in the tank, the more pronounced the problem is. Is it possible that a full tank prevents the clogging? What do you think?
You may need a new fuel pump.
With the engine off and ignition off and parked in a quiet place which I know you have, turn the ignition on and listen to the fuel pump hum in the rear seat area (in the fuel tank). It should sound even pitched and then turn off after a couple of seconds once the fuel line is primed. If it either goes on and on or not at all then you may likely have a faulty pump. You can also disconnect the fuel line at the engine and judge how smoothly the fuel comes out with the ignition on. If it is slow or erratic then try the same under the car before the fuel filter and before the longest part of the fuel line, which can tell you if you have any unlikely blockage in the line. Again if the supply is weak then you will likely have a pump problem.
If the fuel checks out then try the coil. A few Ventos were said to suffer from coil problems which would sometimes give the impression of a faulty fuel pump.
Thanks for the help, but if it were a fuel pump problem or a coil problem, wouldn’t it be constantly erratic and not just when my fuel is low?
Sorry, missed the bit about being only when low on fuel. [my brain is foggy today. Must be the humidity!]
Try taking off the petrol/gasoline filler cap when you are having issues. If idle returns to normal then you probably have excessive vacuum in the tank which is restricting fuel flow.
It is possible, depending on the system you have that there is an evaporation purge canister fitted to your vehicle (I’m not sure about your car) and that the valve is stuck or not operating properly. It may need replacement, otherwise your carbon canister may require replacement if it has broken down and is restricting gas flow.
If you don’t have one of these fitted then you may have a vacuum valve or breather fitted to the tank itself which may be faulty, (although I believe your car should have a evap can fitted)
You could probably just pull the damn thing off your car and not notice, but I didn’t tell you that as this sort of advice is surely environmentally unfriendly. 
It may also be slightly possible that your fuel pick-up pipe/filter has dropped off the end of the fuel pump, but I’d check the vacuum status first before you go pulling things to bits.
You have water in your tank…
you have a fuel problem. or maybe an ignition problem. or maybe just a temperamental car.
You’d think so but, I had the same issue with my car and it turned out to be the fuel pump. When the tank is full there is enough fluid head that the pump doens’t have to work as hard.
Having said all of this I wouldn’t discount any of the other suggestions. I’m just going by my own experience.
I’m going to bet it’s rust from the fuel tank. Have a look at the petrol filter and see if you can see little tiny bits in it.
I had an old Golf that had the same problem.
Incidentally, at a different time it also had that coil problem that was mentioned previously, but that had different symptoms