Is it possible to create an external electric field using a toroid

You dont want to shield a compass, then it wont show you the north at all. What you might want to do is make some low pass filtering to get rid of noise. For example avarage the reading over n samples, then the random noise from environment mostly cancels out. Same effect as frictional dampening in old school compass, electronic compass doesnt have any so add it in code.

Also dont forget that compass never points to true north or even magnetic north.

In aeronavigation 3 correction parameters are used:

1. Correction for airplane frame, these are calibration values gotten by rotating airplane and comparing compass readings to magnetic north. This parameter is added to reading.

2. Correction for local magnetic anomalies, due to uneven distribution of metals in earth crust compass reading rarely points straight to magnetic north. There are maps for this where a correction parameter is taken from and added to reading.

3. Correction for geographic coordinates, magnetic north pole is not at north pole, depending on your coordinates a correction parameter must be added, for example if you were flying right between magnetic north and geographic north the correction parameter would be 180degrees. The parameter is less drastic further away you are from poles.

Sum of these 3 parameters and compass reading is geographic north. So you can see that answering "where is north?" with a compass can be a quite difficult if you want high degree of accuracy. If you dont want to count for all this settle for "north is in that generic direction"

Not having used said app I believe it is a master of assumptions and averages.

The far field of a transmission line with two opposite current conductors is essentially zero. However the near field on anything other than a solid screen co-ax will always have a unbalanced component around it.

The maximum unbalance will be when the two current conductors are 'eclipsed' and the near field will be mostly due to the nearer conductor.

What I suggest for better trials is that you take a mains cable and carefully remove the outer covering for 12 inches or so to separate the internal wires (do this with a spare kettle cord if you like). Then run your tests with the phone placed next to just one of the live wires and see if it has repeatable results.

It is also important to remember that the magnetic field sensor in the phone should be a 3 axis device so it can determine the magnetic field 'direction' and infer the conductor placement to a degree and correct a bit.

Accuracy of 30% compared to a somewhat more predictable device I think is pretty phenomenal like you have found. Getting better will obviously be much harder given the loose test conditions required.

It does remind me of a totally passive DC ammeter that my friend had that was over 50 years old. It was merely placed against a (single) conductor (such as the alternator charging lead in a car) and would deflect its needle to indicate the local induced magnetic field proportional to the current. Orientation was important and it would not have read accurately if a return conductor was nearby, however the principal is pretty much the same.