Simulated completeness of the RGB+BH binary catalogue as a function of BH mass and orbital period. Top panels: Mean values of predicted ruwe (left) and rv_amplitude_robust (right) from mock Gaia DR3 observations, shown for simulated binaries with BH masses of 5, 10, and 20 M_⊙. The shaded regions indicate one standard deviation from the mean. The threshold values for the cuts on excess scatter in RV and the position applied to the candidate sample are shown in black. Higher BH masses generally yield stronger astrometric and RV signals. The astrometric ruwe increases with orbital period up to the DR3 baseline (∼1000 d), while the RV amplitude is highest for short-period systems. Bottom panels: Recovery fractions for simulated binaries that pass the initial cuts on excess scatter in RV and position (left), that exceed the inferred companion mass threshold of 3 M_⊙ (centre), and that meet both criteria (right). The final completeness is highest (≳80% for 10–20 M_⊙ BH) for binaries with periods of 100–2000 d. Completeness decreases for longer periods due to incomplete phase coverage within the DR3 observing baseline. For reference, we show approximate completeness fractions for the Gaia DR3 astrometric (dashed) and spectroscopic (dotted) orbital solutions estimated based on the gaiamock forward model and SBX catalogue of spectroscopic binaries (see Sect. 4.6 for details).