This is a rather theoretical question, but is it possible in physics to have a description of reality which does not consist of particles but only fields. Or are fields always defined in such a way that particles must exist?

The problem I think is that at the very fundamental level of physics, we have matter/particles defined in terms of spacetime, and spacetime defined in terms of matter/particles. You can not define one without the other.

To measure something in empty space is impossible, and likewise, to define particles without spacetime is also impossible.

Fields are created by charges and particles. For example, the Gluon and the Color Charge create the color field, which is the strong nuclear force.

And QED and QCD says particles are a result of their respective fields.

From a QM point of view, particles are necessary, but not for many properties. Not necessarily for being points, but to count them, and as a nice way to say that whan the extension of a particle changes, some properties keep conserved, for instance the charge. Also, to avoid that a particle repels itself.