No because light-speed is the speed of causality. It is by all definitions, real time
Due to this relative motion we will see different ordering of events at different locations, despite both being at rest relative to the CMW background, so we can’t really say there is a global timeline.
The thing is that the CMW background isn’t global, we don’t see the same background as a distant observer. CMW photons that are passing us now will take time to reach a distant observer, and vice versa.