Stalls out when backing off the accelerator

This sounds exactly like what happened to my '88 Grand Caravan a few years ago, and you're right about it being a fuel pressure problem. In my case it was caused by a pickup screen in the tank plugged with a rust-colored mud. Now that I know what happened, a clue would have been to unplug the vacuum hose going to the fuel pressure regulator and plug it. That will raise the pressure and possibly cause some minor black smoke from the tail pipe, but it should stop or reduce the stalling.

What happens is due to the blockage, not enough fuel VOLUME can be pumped when the highest volume is needed, ... Which is during coasting. Pressure is reduced when intake manifold vacuum goes up during coasting. That prevents an overly rich coast-down condition. Due to the lower pressure, meaning it's easier for fuel to flow through the regulator back to the tank, the volume goes up. With the higher volume, the blockage causes a pressure drop that is too great, hence the stalling.

I understand "nuts". Other than diesel trucks, you will never solve a running problem on a Chrysler product by replacing the fuel filter so now they also build them inside the tank. I don't know much about Mazda fuel supply systems but I replied because your dandy description matched my observations perfectly. I looked up the various parts on rockauto. Com and found your system does have a pressure regulator, (they call it a damper), and it has a vacuum port. That regulator is now built into the pump assembly on a lot of cars and sits inside the tank. I'm not sure where yours sits so there is either going to be a long vacuum hose going to the tank or the regulator will be on the fuel rail on the engine.

One interesting note is they list a fuel pressure sensor. On my van, and on most cars, fuel pressure is not monitored so there will be no diagnostic fault code related to pressure. All you get are codes related to the RESULTS of incorrect pressure such as "running too rich too long" or "running too lean too long". I don't know what is done with that sensor reading. The Engine Computer could use it to calculate how long to pulse the injectors which controls how much fuel enters the engine, or it might just be to keep tabs on the pressure in case it goes too high or too low.

For what it's worth, they list the strainer separately, and it looks exactly like what's attached to my pump housing.