We use self-consistent numerical simulations of the evolution and disruption of the Sun's birth cluster in the Milky Way potential to investigate the present-day phase-space distribution of the Sun's siblings. The simulations include the gravitational N-body forces within the cluster and the effects of stellar evolution on the cluster population. In addition, the gravitational forces due to the Milky Way potential are accounted for in a self-consistent manner. Our aim is to understand how the astrometric and radial velocity data from the Gaia mission can be used to pre-select solar sibling candidates. We vary the initial conditions of the Sun's birth cluster, as well as the parameters of the Galactic potential. In particular, we use different configurations and strengths of the bar and spiral arms. We show that the disruption time-scales of the cluster are insensitive to the details of the non-axisymmetric components of the Milky Way model and we make predictions, averaged over the different simulated possibilities, about the number of solar siblings that should appear in surveys such as Gaia or GALAH. We find a large variety of present-day phase-space distributions of solar siblings, which depend on the cluster initial conditions and the Milky Way model parameters. We show that nevertheless robust predictions can be made about the location of the solar siblings in the space of parallaxes (ϖ), proper motions (μ) and radial velocities (Vr). By calculating the ratio of the number of simulated solar siblings to that of the number of stars in a model Galactic disc, we find that this ratio is above 0.5 in the region given by: ϖ ≥ 5 mas, 4 ≤ μ ≤ 6 mas yr−1, and −2 ≤ Vr ≤ 0 km s−1. Selecting stars from this region should increase the probability of success in identifying solar siblings through follow-up observations. However the proposed pre-selection criterion is sensitive to our assumptions, in particular about the Galactic potential. Using a more realistic potential (e.g. including transient spiral structure and molecular clouds) would make the pre-selection of solar sibling candidates based on astrometric and radial velocity data very inefficient. This reinforces the need for large-scale surveys to determine precise astrophysical properties of stars, in particular their ages and chemical abundances, if we want to identify the solar family.