Reality courtroom series have several permutations.
They may be real cases with real judges who hand down a verdict to which the parties must adhere. They may have real judges, while actors portray the attorneys and/or parties to a case which either has already taken place or is fictional. Historical cases may be completely re-enacted. In the most far-fetched stretch of the genre, a game show has been organized on courtroom conventions.
Reality Courtroom Series: 1949-2005, organized by title, with series descriptions (PDF)
Reality Courtroom Series: 1949-2005, organized by date (PDF)