reciprocal SAW devices, in spite of the fact that pure acoustic waves in solids are reciprocal. [4][5][6] Although nonreciprocity of magnetoelastic waves (e.g., hybridized SAW-SW) have been studied for a long time, [7,8] it is only recently the practically usable levels (giant and/or ultra-wideband) of nonreciprocity of SAW propagation losses have been achieved. [9][10][11][12][13][14] These novel results open a way for the development of efficient microwave SAW isolators, compatible with the modern SAW technology.Isolators, however, are not the only nonreciprocal devices, that are necessary for the further development of microwavefrequency SAW-based signal processing technology. Circulators and nonreciprocal phase shifters are also needed.Although there exists a method to build a circulator using several isolators, [15,16] this method is hardly compatible with the existing SAW technology. Indeed, the realization of an isolator-based circulator requires the utilization of the Wilkinson-type power splitters, which are known for electromagnetic waves, but have not been realized (and even theoretically studied) for SAWs. It is clear, that additional transformations from SAW to electromagnetic wave and back eliminate almost all the advantages of the SAW-based devices.A standard circulator scheme, used, in particular, in electromagnetic [17] and ultrasonic [18] devices, relies, in contrast, on the phase nonreciprocity of waves, when the phase accumulation for waves propagating in opposite direction is not the same. This phase effect is, of course, caused by the nonreciprocity of the wave dispersion relation. As it is explained below, it is not trivial to achieve the practically useful levels of phase nonreciprocity of a hybridized magnetoelastic wave, because, usually, the strongly hybridized acoustic waves have prohibitively large losses, that are much larger than the losses of pure SAW. In this work we demonstrate, how this problem can be solved by the utilization of magnetoelastic coupling between SAW and highly nonreciprocal SWs, which could exist, for instance, in a synthetic antiferromagnet (SAFM), and discuss possible realization of nonreciprocal SAW phase shifters and circulators.