Media Foundation workarounds are required to play pre-rendered cutscenes. This isn't very complicated, but it will require you to go into Desktop mode on your Deck, and unfortunately the documentation is pretty spartan. The developer of mf-install thinks you're an idiot if you can't solve this ("It's Computing 101 stuff"), so I'll provide more in-depth instructions.

First, you'll want to go to https://github.com/z0z0z/mf-install and download the files. If you're not familiar with Git, the fastest way to do this is to press the big green "Code" button and click "Download ZIP." Extract the ZIP somewhere and enter the folder for it, then open a terminal window in that folder. You'll need to run the following command:

WINEPREFIX="/home/deck/.steam/steam/steamapps/compatdata/889600/pfx" PROTON="/home/deck/.steam/steam/steamapps/common/Proton 7.0" ./mf-install.sh -proton

After that, you can go back to Gaming Mode, and it should run fine. More explanation in the concluding notes.

If you're not familiar with the command line, here's a run-down of what that command does, in case you need it for another game. The first part, WINEPREFIX="filepath", tells the script what game you're trying to install to. The first part of the filepath is just where your steam folder lives; the numbers are the end are the steam ID for The Caligula Effect. The second part, PROTON="filepath", tells it which version of Proton you're trying to install to. I used Proton 7.0.1 here. The rest of the command just runs the script. You need to run this from the folder that you extracted the script to. This is all pretty straightforward for someone with CS experience, but it can be pretty daunting for a novice. Hopefully this helps!