Spin-wave based devices offer several advantages, such as the absence of Joule losses and the sub-μm wavelength in the GHz-THz range, and have been proposed as promising alternatives to the standard CMOS technology. In this context, synthetic antiferromagnetic systems have been extensively studied for the development of nanomagnonic devices, thanks to their high degree of tunability. Moreover, spin textures have recently been demonstrated as efficient means for the generation and emission of spin waves. Here, we show that with the newly proposed phase nanoengineering methodology it is possible to magnetically nanopattern spin textures via thermally assisted magnetic Scanning Probe Lithography in a 200 nm thick exchange-biased synthetic antiferromagnetic multilayer. In such nanopatterned structures, we demonstrate via time-resolved Scanning Transmission X-Ray Microscopy the generation and manipulation of different types of coherent spin-wave modes. By strongly enhancing the robustness and quality of the spin-wave wavefronts propagating for multiple wavelengths in thick synthetic antiferromagnetic systems, this work opens the possibility to expand the comprehension of the spin-wave phenomenology also to the third dimension and to study the complex spin-wave properties through the volume of the magnetic systems, enabling their control for the design of novel three-dimensional nanomagnonic devices.