Each of the studied manifolds has a pair of B-metrics, interrelated by an almost contact structure. The case where each of these metrics gives rise to an η-Ricci–Bourguignon almost soliton, where η is the contact form, is studied. In addition, the geometry-rich case where the soliton potential is torse-forming and is pointwise collinear on the Reeb vector field with respect to each of the two metrics is considered. Ricci tensors and scalar curvatures are expressed as functions of the parameters of the pair of almost solitons. Particular attention is paid to the special case when the manifold belongs to the only possible basic class of the corresponding classification. A necessary and sufficient condition has been found for these almost solitons to be η-Einstein for both metrics.