  We present solutions of force-free pulsar magnetospheres coupled with a
uniform external magnetic field aligned with the dipole magnetic moment of the
pulsar. The inclusion of the uniform magnetic field has the following
consequences: The equatorial current sheet is truncated to a finite length, the
fraction of field lines that are open increases, and the open field lines are
confined within a cylindrical surface instead of becoming radial. A strong
external magnetic field antiparallel to the dipole allows for solutions where
the pulsar magnetic field is fully enclosed within an ellipsoidal surface.
Configurations of fully enclosed or confined magnetospheres may appear in a
double neutron star (DNS) where one of the components is a slowly rotating,
strongly magnetised pulsar and the other a weakly magnetised millisecond
pulsar. Depending on the geometry, twisted field lines from the millisecond
pulsar could form an asymmetric wind. More dramatic consequences could appear
closer to merger: there, the neutron star with the weaker magnetic field may
undergo a stage where it alternates between an open and a fully enclosed
magnetosphere releasing up $10^{38}$erg. Under favourable combinations of
magnetic fields DNSs can spend up to 10$^3$ yr in the coupled phase, implying a
Galactic population of $0.02$ systems. Potential observational signatures of
this interaction are discussed including the possibility of powering recurring
Fast Radio Bursts (FRBs). We also note that the magnetic interaction cannot
have any significant effect on the spin evolution of the two pulsars.
