The title pretty much says it all. It is widely taught that a gene in a eukaryotic system could produce more than one protein due to post-transcriptional modification, but I do not believe I have come across any specific examples of this. Are any such systems known? Or is this more theoretical?
There are a large number of ways a protein variant can be produced by post translational modification. The question may seem obvious, but its really quite broad.
I can start this out. I doubt I know all the ways a single transcript can produce variant proteins. A detailed description might be more like a review article than an answer here.
First, there are very rare cases of eukaryotic polycistronic mRNA: mRNA which code for more than one protein. In this case, after translation, the mRNA are processed to make multiple monocistronic RNA.
Then there are proteins with molecules added by covalent bonding to the protein. Glycoproteins are notoriously variable. Many other small molecules can be bonded to a protein to make a variant of this sort: ubiquitinization, N-acetylation, SUMO domains, lysine methylation are just a few of the more common covalent modifications of proteins.
There are RNA editing which removes some of the coding RNA to make alternative splice RNA.
The list goes on and on. So does any of this come close to what you wanted to know?
The answer is not simple - @shigeta mentioned a few mechanisms leading to single gene-to-multi protein relationships - and the answer is certainly not short (there are thousands of these genes).
But anyway "alternative splicing" seems to be the primary mechanism according to this article, so rather than listing all alternatively splicing genes, here are the databases (+links):
Alternative splicing gene database (click on blue for link):
EDAS: EST derived alternative splicing database
U12DB: A database of U12 spliceosomal introns
FASTdb: Friendly Alternative Splicing and Transcript database
BIPASS: Bioinformatics Pipeline for Alternative Splicing
ASAP II: Alternative Splicing Database
ASTD: Alternative Splicing and Transcript Diversity database
H-DBAS: Human database of alternative splicing
Hollywood: Alternatively spliced mRNA database
Ecgene: Genome annotation for alternative splicing
SpliceMiner: Collection of splice variants for human genes
Just for balance, here is an example of a single protein being constructed from two primary gene products (two separate genes involved) via protein splicing.