The lines of the earth’s magnetic field are approximately parallel to the earth’s surface at the equator, and dive into the earth near the magnetic north/south poles. If you measure the local incline of the field lines with respect to gravity, is it possible to determine your latitude with any useful degree of accuracy (say, ± 5 degrees of latitude)?
It’s called an “inclination compass” and it’s part of the mechanism that various migratory animals use for navigation. It can be demonstrated by observing animal behavior when you apply various external magnetic fields: there are animals that will orient themselves to the magnetic inclination, regardless of the polarity of the field.
The precise mechanism is still rather mysterious, though it has something to do with cryptochrome proteins. This is a light-sensitive protein that’s found in the eyes of birds and insects (which means the inclination compass only works in daylight), but somehow the light-activated reactions are sensitive to very weak magnetic fields. Fruit flies that have do not have cryptochromes are behaviorally insensitive to magnetic fields.
ETA: For reference, the review I used to jog my memory past “what was all that business with the cryptochromes and magnets again?” is here.
The navigator’s term of art for this is “magnetic dip”. See Magnetic dip - Wikipedia for a good article and for a map of the values around the world.
You’ll see that you already need to know your longitude because the dip values are way off being parallel to the lines of latitude. Absent longitude you really can’t do much position determination just from the dip angle. Given longitude it’d be an easy table lookup (but see below) to convert dip angle to latitude with reasonable precision.
You’ll also see there are some areas of the world where it’d work slick and other areas where it’d work terribly. Most of the really bad areas are in the deep southern latitudes.
Just for the record, latitude is the one that’s easy to measure (almost any astronomical observation at all will do it). Longitude is usually done by combining an astronomical measurement with an accurate clock.
A dip compass could get you an approximate value for “magnetic latitude”, or distance from the magnetic pole, and that in combination with your real latitude could in principle give you your location. But the magnetic latitude will be iffy because of local irregularities in the field, and the calculation is such that small errors in the inputs will be magnified into large errors in your position.