Euclid Data Release 1 (DR1). Colouring Euclid with external ground-based imaging data

Accurate photometric-redshift estimation and calibration of the weak-lensing point-spread function (PSF) for Euclid require complementary ground-based optical imaging with high photometric precision and spatial homogeneity. The Organisation Unit for External Data (OU-EXT) integrates heterogeneous ground-based surveys into the Euclid Science Ground Segment (SGS), delivering uniformly calibrated, co-added images, and associated PSF models. To minimise systematics in spectral energy distribution (SED) reconstruction and photometric-redshift estimation, these data must achieve relative photometric stability at the ~10 mmag level on Euclid-tile scales. We present the ground-based imaging component of Euclid Data Release 1 (DR1), comprising ~2500 deg² across the wide, deep, and auxiliary fields. We summarise the OU-EXT production workflow and validation framework, and quantify the achieved data quality of the co-added products in terms of depth, photometric zero-point stability, and spatial colour homogeneity. In the Euclid Wide Survey (EWS), the point-source $10σ$ depths in the $griz$ bands reach 24.8/24.3/24.0/23.0 AB mag, with the deep fields extending 1--2 magnitudes deeper. Most EWS DR1 tiles meet the 10 mmag relative photometric requirement, with 1.1% and 1.5% showing offsets exceeding 50 mmag in the northern and southern hemispheres, respectively. Spatial colour variations remain typically below 5--10 mmag across the southern EWS footprint, while larger residual structures of 10--25 mmag appear in some northern regions. These results demonstrate that the EXT production chain provides a largely homogeneous framework for integrating ground-based datasets into Euclid, while identifying calibration and co-addition limitations that have to be addressed in DR2 to meet the mission's most stringent cosmological requirements. (Abridged)