Pair potentials that are bounded at the origin provide an accurate description of the effective interaction for many systems of dissolved soft macromolecules (e.g., flexible dendrimers). Using numerical free-energy calculations, we reconstruct the equilibrium phase diagram of a system of particles interacting through a potential that brings together a Gaussian repulsion with a much weaker Gaussian attraction, close to the thermodynamic stability threshold. Compared to the purely repulsive model, only the reentrant branch of the melting line survives, since for lower densities solidification is overridden by liquid-vapor separation. As a result, the phase diagram of the system recalls that of water up to moderate (i.e., a few tens of MPa) pressures. Upon superimposing a suitable hard core on the double-Gaussian potential, a further transition to a more compact solid phase is induced at high pressure, which might be regarded as the analog of the ice I-to-ice III transition in water.