I think @adrianlison has captured the generative process version of this problem quite well and agree it seems like quite a hard nut to crack. @johannes suggestion of starting with a distribution with negative support seems like a good one as a first pass and agree this differs from negative delays (though solutions that help with either will likely help with the other to some extent).
@johannes in your case what is the typical delay for negative updates and how often do they happen? I think there is another version of this problem where there is no real distribution and you instead have irregular large-scale system changes that lead to updates across the time series. Not a nowcasting time-series but the behaviour of reported COVID-19 in the JHU data set comes to mind as an example of this. I haven’t seen much on dealing with this (and actually thinking of writing a short note) but on the face of it that problem may be almost impossible to model.
Just to expand on @teojcryan project (more detail: Nowcasting COVID-19 cases by specimen date in England).
The basic issue was to nowcasting COVID-19 cases by specimen data in the UK. The problem (as @teojryan highlights) was that case confirmation can come from either an LFT test or a PCR test. When a LFT test is used a PCR test is commonly used to check. Sometimes the LFT is a false positive and so the PCR test leads to the case being removed (regardless of what happens the first date available for a case is kept). This leads to a dynamic where cases are truncated by some distribution (a mixture of LFT and PCR test to report delays) and then updated based on another distribution (the time from LFT to follow PCR) which has no effect when the PCR is positive. As @adrianlison points out a sensible way to model this would be as two separate processes. As he also points out this could very well be quite hard to identify (though there is information there so likely not impossible).