Real three-dimensional biquadrics

We find the topological types of biquadrics (complete intersections of two real four-dimensional quadrics). The rigid isotopy classes of real three-dimensional biquadrics were described long ago: there are nine such classes. We find the correspondence between the topological types of real biquadrics and their rigid isotopy classes, and show that only two rigid isotopy classes have the same topological type. One of these classes consists of real $\operatorname{GM}$-varieties and the other contains no $\operatorname{GM}$-varieties. We also study the sets of real lines on real biquadrics.