Summary: We first study coactions of dual quasi-bialgebras or dual quasi-Hopf algebras. If \(H\) is a dual quasi-Hopf algebra and \(C\) is a coalgebra in the tensor category of left \(H\)-comodules \(^H{\mathcal M}\), we define the smash coproduct \(C\rtimes H\) by a universal property and then we prove that it has a realization on \(C\otimes H\). Second, since \(C\rtimes H\) is an ordinary coalgebra, we study the categories of left (right) \(C\rtimes H\)-comodules. In fact, we prove that if \(C\) or \(H\) is finite-dimensional, then these categories are isomorphic to the categories of relative Hopf modules introduced in our earlier paper [J. Algebra 229, No. 2, 632-659 (2000; Zbl 0964.16035)].