Poetic justice is when justice is carried out with poetry. In court, for example, the defendant, plaintiff, justice, lawyers and jury all have to perform the whole trial in poetry or in rhyming prose. Blank verse is not permitted.
This can be done at the defendant's request (and amusement). Or, if the justice hearing the trial wishes it, whether or not the defendant wants it.
The best poetic justices and lawyers are Vogons as their poetry, even in judicial language, is the 33rd best poetry in the galaxy. Appeals are rare, as appellate court judges can rarely bear to read the transcripts. You see, the poetry judges are all Vogons, and a Vogon's idea of what is good poetry is unlike that of any other member of any other race in the multiverse.
Frankly, almost none of this is true. But I'm not telling what.