Evolved low to intermediate mass stars are known to shed their gaseous envelope into a large dusty molecule-rich circumstellar nebula which typically develop a high degree of structural complexity. Most of the large-scale, spatially correlated structures in the nebula are thought to originate from the interaction of the stellar wind with a companion. As part of the Atomium large programme, we observed the M-type asymptotic giant branch star R Hydrae with the Atacama Large Millimeter/submillimeter Array (ALMA). The morphology of the inner wind of R Hya — which has a known companion at ∼3500 au — was determined from maps of CO and SiO obtained at high angular resolution. A map of the CO emission reveals a multi-layered structure consisting of a large elliptical feature at an angular scale of ∼10”, that is oriented along the North–South axis. The wind morphology within the elliptical feature is dominated by two hollow bubbles. The bubbles are on opposite sides of the AGB star and lie along an axis with a position angle of ∼115◦. Both bubbles are offset from the central star, and its appearance in the SiO channel maps indicates that the two bubbles might be shock waves travelling through the AGB wind. An estimate of the dynamical age of the bubbles yields an age of the order of a hundred years, which is in agreement with the previously proposed elapsed time since the star last underwent a thermal pulse. When the CO and SiO emission is examined on sub-arcsecond angular scales, there is evidence for an inclined, differentially rotating equatorial density enhancement, strongly suggesting the presence of a second nearby companion. The major axis of the position angle of this disk is ∼70◦, in the plane of the sky. We tentatively estimate that a lower limit on the mass of the nearby companion is ∼0.65 M⊙, on the basis of the highest measured speeds in the disk and the location of its inner rim at ∼6 au from the AGB star.