So you bought an altitude-azimuth mount[a] and now you're sad because while it will track stars, there is a field rotation effect that results in star trails around the image center. Fear not. You can convert it to an equatorial mount[b] with a tripod.
Table of Contents
Place the mount on a tripod and tilt it so that the azimuth axis is parallel to the Earth's rotational axis. Then the mount only has to rotate around that axis to completely compensate for the Earth's rotation, and you get no field rotation.
2. Axis Alignment
The amount to tip depends on your latitude and toward which pole you tilt. First, figure out which latitude you are on. Then tip 90 minus the number of degrees as your latitude toward the nearest pole. If you live in London (latitude 51 degrees north), tip the mount 39 (90 minus 51) degrees north. If you live in Ushuaia (latitude 55 degrees south), tip the mount 35 (90 minus 55) degrees south.
For example, if you, like me, live in Stockholm, Sweden, at 59.4 degrees northern latitude - tilt the mount 30.6 degrees toward north, and your azimuth axis is now aligned with the Earth's.
2.1. On the Equator
As can be seen from the above, the tilt gets worse and worse the closer to the equator you are. Unfortunately there is no way to "tip toward the equator": since the altitude axis itself rotates when the mount rotates around the azimuth axis, we can't use the altitude axis for alignment.
Finally, you must tell the mount that you are at one of the poles: When asked for observation position, enter 0 degrees east and either 90 degrees north or south.
4. Auto-Tracking (AZ Goto)
The AZ Goto mount has an auto-tracking feature. This section is specific to that particular model, but other mounts probably have similar requirements.
In order for auto-tracking to work, the mount must start off pointed north so the mount can figure out which direction to turn the axis. This means that it is easiest to tip the mount "backwards" when tipping it toward the south pole. Once set up, go to Setup - Auto Tracking and press Enter. Then use the up and down menu selection keys (at the bottom of the handset) to switch position display and confirm that the mount is tracking. If you slew the mount with auto-tracking on, the mount will resume tracking if you press Enter.
5. Test Results
These photos were taken with the setup described above. I eyeballed north, tilted the mount roughly 30 degrees north and took some sample shots at 20 seconds exposure. Unfortunately the weather was near overcast, which meant that I had to photograph through holes in the clouds. I did get some stars, though, and I think the results at least validate the setup.