Multiplicity is one of the most fundamental observable properties of massive O-type stars and offers a promising way to discriminate between massive star formation theories. Nevertheless, companions at separations between 1 and 100 mas remain mostly unknown due to intrinsic observational limitations. [...] The Southern MAssive Stars at High angular resolution survey (SMASH+) was designed to fill this gap by providing the first systematic interferometric survey of Galactic massive stars. We observed 117 O-type stars with VLTI/PIONIER and 162 O-type stars with NACO/SAM, respectively probing the separation ranges 1-45 and 30-250mas and brightness contrasts of Delta H < 4 and Delta H < 5. Taking advantage of NACOs field-of-view, we further uniformly searched for visual companions in an 8-radius down to Delta H = 8. This paper describes the observations and data analysis, reports the discovery of almost 200 new companions in the separation range from 1mas to 8 and presents the catalog of detections, including the first resolved measurements of over a dozen known long-period spectroscopic binaries. Excluding known runaway stars for which no companions are detected, 96 objects in our main sample (DEC < 0 deg; H<7.5) were observed both with PIONIER and NACO/SAM. The fraction of these stars with at least one resolved companion within 200mas is 0.53. Accounting for known but unresolved spectroscopic or eclipsing companions, the multiplicity fraction at separation < 8 increases to f_m = 0.91 +/- 0.03. The fraction of luminosity class V stars that have a bound companion reaches 100% at 30mas while their average number of physically connected companions within 8 is f_c = 2.2 +/- 0.3. This demonstrates that massive stars form nearly exclusively in multiple systems. Additionally, the nine non-thermal (NT) radio emitters observed by SMASH+ are all resolved [...]
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