Antiferromagnetically coupled multilayers with perpendicular anisotropy, as [CoPt]/Ru, Co/Ir, Fe/Au, display ferromagnetic stripe phases as the ground states. It is theoretically shown that the antiferromagnetic interlayer exchange causes a relative shift of domains in adjacent layers. This "exchange shift" is responsible for several recently observed effects: an anomalous broadening of domain walls, the formation of so-called "tiger-tail" patterns, and a "mixed state" of antiferromagnetic and ferromagnetic domains in [CoPt]/Ru multilayers. The derived analitical relations between the values of the shift and the strength of antiferromagnetic coupling provide an effective method for a quantitative determination of the interlayer exchange interactions. [2,3,4,5]. Due to the strong competition between antiferromagnetic interlayer exchange and magnetostatic couplings [3,6], nanoscale superlattices with strong perpendicular anisotropy display specific multidomain states and unusual magnetization processes [2,3,5,7], which have no counterpart in other layered systems with perpendicular magnetization [8].So far, theoretical analysis of magnetization states and processes in antiferromagnetically coupled multilayers with out-of-plane magnetization has been based on micromagnetic models of stripe domains, where the domain walls throughout the whole stack of the ferromagnetic layers sit exactly on top of each other [3,6]. In our letter we show that this assumption is wrong. The antiferromagnetic interlayer coupling causes a lateral shift of the domain walls in the adjacent ferromagnetic layers. We develop a phenomenological theory of these complex stripe states. The analytical evaluation of a basic twolayer model shows that the formation and evolution of such "shifted" multidomain phases should appreciably influence the appearance and the magnetization processes of stripe states in perpendicular, antiferromagnetically coupled multilayers.As a model we consider stripe domains in a superlattice composed of N identical layers of thickness h antiferromagnetically coupled via a spacer of thickness s. The stripe domain phase consists of domains with alternate magnetization M along the z-axis perpendicular to the multilayer plane. The domains are separated by thin domain walls with a finite area energy density σ. The magnetic energy density of the model (Fig. 1 (a) ) can be written as a function of the stripe period D and the shift aThe first term in (1) describes the domain wall energy, w m is the stray field energy, J > 0 is the antiferromagnetic exchange interaction. The upper (lower) sign corresponds to an (anti)parallel arrangement of the magnetization in the adjacent layers. We call these modes ferro and antiferro stripe phases. We introduce a set of reduced geometrical parametersand two characteristic lengthsdescribing the relative energy contributions of the domain walls (l) and the interlayer coupling (δ) in comparison to the stray field energy. Then, the reduced energy w = W/(2πM 2 N ) can be writtenThe stray field ene...