Is there a translational mechanism that eukaryotes can use to produce different proteins from a single transcribed mRNA?
There are multiple mechanisms that are known to lead to translation of substantially (or entirely) different proteins from a single mRNA. While these mechanisms are more typically seen in viruses, I'm focusing on examples documented within the endogenous transcriptome of eukaryotes.
Alternative translation initiationOne process that can lead to different proteins being translated from the same mRNA in eukaryotes is the use of alternative translation initiation sites.1,2 Translation typically starts with a [pre-initiation complex](https://en.wikipedia.org/wiki/43S_preinitiation_complex) recognizing the 5' cap and loading onto the mRNA. This complex scans until it finds an appropriate start site. The choice of start site depends on the how well the ribonucleotide sequence matches the [Kozak consensus sequence](https://en.wikipedia.org/wiki/Kozak_consensus_sequence).3 If the region around the first AUG is not a good match for that consensus then a process known as [leaky scanning](https://en.wikipedia.org/wiki/Leaky_scanning) occurs and the pre-initiation complex can continue along the mRNA until a "good" start site is found.
While this can result in proteins with different functions, they will typically still have large regions of amino acid sequence in common. One example of this is in a kinase known as MK2, which is "a key regulator of transcription, migration, death signaling and post-transcriptional gene regulation".4
Polycistronic mRNAsWhile more common in prokaryotes, in some cases eukaryotes also have polycistronic transcripts5,6. These transcripts encode multiple separate proteins (i.e. from independent open reading frames). The examples that I have found for mammals are all bicistronic (operons with two genes): LASS1-GDF1, SNRPN-SNURF, MTPN-LUZP6 and MFRP-C1QTNF5. You can search for those gene pairs, but there doesn't seem to be a huge amount of information available and in many cases one of the genes is almost completely uncharacterized. Eukaryotic operons (aka polycistronic mRNAs) are ubiquitous in trypanosomes, appear to be very common in nematodes (round worms), and are also frequently seen in Drosophila (a fly).6
Translational read-through[Translational readthrough](https://en.wikipedia.org/wiki/Stop_codon#Translational_readthrough) aka. stop codon suppression occurs when an in frame stop codon is ignored either stochastically or under specific conditions and translation continues beyond that point. This leads to a C-terminal extension of the protein which in some cases has been shown to have functional implications7,8. An example of this is the [PSI+] prion in yeast, which promotes translational readthrough throughout the yeast transcriptome by inactivating a factor involved in translation termination.7
Translational frameshiftsThis is well covered in the answer by @acvill.
3: Acevedo, J. M., Hoermann, B., Schlimbach, T., & Teleman, A. A. (2018). Changes in global translation elongation or initiation rates shape the proteome via the Kozak sequence. Scientific reports, 8(1), 4018. 4: Trulley, P., Snieckute, G., Bekker-Jensen, D., Menon, M. B., Freund, R., Kotlyarov, A., ... & Gaestel, M. (2019). Alternative Translation Initiation Generates a Functionally Distinct Isoform of the Stress-Activated Protein Kinase MK2. Cell reports, 27(10), 2859-2870.
8: Loughran, G., Jungreis, I., Tzani, I., Power, M., Dmitriev, R. I., Ivanov, I. P., ... & Atkins, J. F. (2018). Stop codon readthrough generates a C-terminally extended variant of the human vitamin D receptor with reduced calcitriol response. Journal of Biological Chemistry, 293(12), 4434-4444.
Separate from the alternative translation start site mechanism that tyersome describes is programmed translational frameshifting. This is independent of alternative splicing or post-translational modifications, and happens when a ribosome switches reading frames while a protein is already being translated. Typically, this phenomenon is associated with viral translation, and allows viruses to encode many proteins on relatively short genomes. Take HIV as an example: the polyprotein gag-pol requires efficient −1 frameshifting for expression of the individual gag and pol gene products.
In eukaryotes, examples are more sparse. Check out this review from 2012 --
... examples of mammalian genes that utilize −1 frameshifting are the mouse embryonic carcinoma differentiation regulated (EDR) gene and its human ortholog PEG10. A slippery sequence of G GGA AAC, in combination with a pseudoknot, mediates highly efficient −1 frameshifting, similar to viral frameshifting motifs (Clark et al., 2007). Recently, a programmed ribosomal −1 frameshift has been identified in the adenomatous polyposis coli (APC) mRNA in Caenorhabditis elegans that is mediated by a slippery sequence A AAA AAA or A AAA AAC (Baranov et al., 2011). The functional relevance of this frameshift is uncertain.
Frameshifting events are often mediated by conserved RNA secondary structures, like pseudoknots and stem-loops. For some specific examples of the types of structures involved in frameshifting, see the following publications:
Some more recent examples of (potential) programmed translational frameshifts in eukaryotes: