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Photonic balls are spheres tens of micrometers in diameter containing assemblies of nanoparticles or nanopores with a spacing comparable to the wavelength of light. When these nanoscale features are disordered, but still correlated, the photonic balls can show structural color with low angle-dependence. Their colors, combined with the ability to add them to a liquid formulation, make photonic balls a promising new type of pigment particle for paints, coatings, and other applications. However, it is challenging to predict the color of materials made from photonic balls, because the sphere geometry and multiple scattering must be accounted for. To address these challenges, we develop a multiscale modeling approach involving Monte Carlo simulations of multiple scattering at two different scales: we simulate multiple scattering and absorption within a photonic ball and then use the results to simulate multiple scattering and absorption in a film of photonic balls. After validating against experimental spectra, we use the model to show that films of photonic balls scatter light in fundamentally different ways than do homogeneous films of nanopores or nanoparticles, because of their increased surface area and refraction at the interfaces of the balls. Both effects tend to sharply reduce color saturation relative to a homogeneous nanostructured film. We show that saturated colors can be achieved by placing an absorber directly in the photonic balls and mitigating surface roughness. With these design rules, we show that photonic-ball films have an advantage over homogeneous nanostructured films: their colors are even less dependent on the angle.