What is the gravitational force between monopoles?

Disclaimer: In this answer we are talking about monopole solutions of classical field theories, rather than about properties monopoles of our universe might have.

Gravitating generalizations of flat space monopoles with Yang–Mills and Higgs fields exist at least for small values of dimensionless gravitational coupling constant (see e.g. this review). At large distances from its core long range fields of an isolated monopole could be described by Reissner–Nordström solution with a positive mass and magnetic charge. So, a pair of such monopoles at large distances would experience gravitational attraction as well as magnetostatic repulsion.

Near the monopole core there is usually a region, where the lapse function of the metric decreases with distance from the center. Such feature could be seen as a region of repulsive gravity in the sense that a point particle placed near the center of the monopole would be repulsed from its core, however if we try to bring two monopoles together this closely, their cores would overlap and it would be impossible to separate gravitational force from the forces of Yang–Mills and Higgs fields.

There is also a class of solutions called global monopoles representing a point-like defects in theories with spontaneously broken global symmetry. (Magnetic monopoles in theories with broken gauge symmetry could then be called “local”). Such solutions when coupled with gravity are not asymptotically flat and possess a solid angle deficit (as well as a small negative effective mass). A pair of such global monopoles would experience gravitational repulsion that does not decrease with the distance (and conversely, a pair monopole–antimonopole would attract each other with almost independent of the distance force). For more details see Barriola & Vilenkin, 1989 and Harari & Loustó, 1990.