It's a problem all right, because the police could arrest someone for littering, find evidence of drug dealing in a phone, and then retroactively claim some suspicion of the latter in order to justify more severe charges.
The technical approach to this would be to clone some existing version control system and have a country's court system run its own equivalent of GitHub, so that police could submit crimes and evidence as they are investigated/discovered but not monkey about with their own records. Of course such a system would not be cost-free, and it would still be open to abuses from both sides.
The version control system you describe was originally designed for text-based code, and would probably impose significant time costs on the officers, and delays on the suspects. Police cameras, which could be required to be turned on at any stop, or before any search/arrest would provide the same information (if the officers were required to state their cause aloud). The cameras could even be set to save 5, 10, or 15 minutes of footage before 'activated', so that the police would not be able to claim they 'were in a rush'. This system could also be automatically activated by a gunshot or other sound.
The technical approach to this would be to clone some existing version control system and have a country's court system run its own equivalent of GitHub, so that police could submit crimes and evidence as they are investigated/discovered but not monkey about with their own records. Of course such a system would not be cost-free, and it would still be open to abuses from both sides.