You are conflating reference frames - it determines causality in your local reference frame but not in any other reference frame. The concept of "real time" - like all time - is entirely relative to the reference frame you are using.
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.