In x-ray computed tomography (CT), task-based image quality studies are typically performed using uniform background phantoms with low-contrast signals. Such studies may have limited clinical relevancy for modern non-linear CT systems due to possible influence of background texture on image quality. The purpose of this study was to design and implement anatomically informed textured phantoms for task-based assessment of low-contrast detection. Liver volumes were segmented from 23 abdominal CT cases. The volumes were characterized in terms of texture features from gray-level co-occurrence and run-length matrices. Using a 3D clustered lumpy background (CLB) model, a fitting technique based on a genetic optimization algorithm was used to find the CLB parameters that were most reflective of the liver textures, accounting for CT system factors of spatial blurring and noise. With the modeled background texture as a guide, a cylinder phantom (165 mm in diameter and 30 mm height) was designed, containing 20 low-contrast spherical signals (6 mm in diameter at targeted contrast levels of ~3.2, 5.2, 7.2, 10, and 14 HU, 4 repeats per signal). The phantom was voxelized and input into a commercial multi-material 3D printer (Object Connex 350), with custom software for voxel-based printing. Using principles of digital half-toning and dithering, the 3D printer was programmed to distribute two base materials (VeroWhite and TangoPlus, nominal voxel size of 42x84x30 microns) to achieve the targeted spatial distribution of x-ray attenuation properties. The phantom was used for task-based image quality assessment of a clinically available iterative reconstruction algorithm (Sinogram Affirmed Iterative Reconstruction, SAFIRE) using a channelized Hotelling observer paradigm. Images of the textured phantom and a corresponding uniform phantom were acquired at six dose levels and observer model performance was estimated for each condition (5 contrasts x 6 doses x 2 reconstructions x 2 backgrounds = 120 total conditions). Based on the observer model results, the dose reduction potential of SAFIRE was computed and compared between the uniform and textured phantom. The dose reduction potential of SAFIRE was found to be 23% based on the uniform phantom and 17% based on the textured phantom. This discrepancy demonstrates the need to consider background texture when assessing non-linear reconstruction algorithms.