The interaction between two shielded monopolar vortices has been investigated experimentally in a nonrotating linearly stratified fluid and by full three-dimensional (3D) numerical simulations. The characteristic Reynolds and Froude numbers in the experiments are approximately Re [[approximate]] 500?1000 and F [[approximate]] 0.2?0.4. In a first set of numerical simulations the flow is initialized with the experimentally obtained two-dimensional horizontal velocity field, which is measured in the horizontal symmetry plane of the vortices, and the lacking initial data such as the density perturbation and the full 3D vorticity field are provided by the so-called diffusion model [Beckers et al., J. Fluid Mech. 433, 1 (2001)]. A comparison of the experimental and numerical data shows that using the diffusion model to supply the full 3D initial data is reliable for the Reynolds and Froude numbers considered in present experiments. Conjecturing a wider range of applicability, a second set of numerical simulations has been performed using initial conditions for the flow field and the density perturbation entirely based on the diffusion model. These numerical simulations enable an investigation of the role of Reynolds numbers (up to Re = 10 000) and Froude numbers (F = 0.30, 0.65, and 1.0) which are outside the experimentally accessible range. These simulations provide a better understanding of the dipole formation process, the influence of the vortex shields and the density perturbation in this process, and of the 3D structure of the dipoles. Compactness of the dipole, entrainment of irrotational fluid from the rear, and tail formation behind the dipole have been discussed. Additionally, experimental and numerical results are reported on the interaction between two shielded monopoles of the same sign in a linearly stratified fluid. It is found that for the full range of Reynolds and Froude numbers mentioned above the presence of the vortex shields prevents the merging of these monopoles.