The cars have an automatic relocker function if you unlock the car but then it doesn't detect a door having been opened it'll re-lock itself. Sounds like that is what your car is doing.
Do the cabin lights work for the door open function; i.e. you have the switch in the centre position so when you open / close a door the lights come on?
If they fail to function (but check the bulbs are working by turning to 'on') then it suggests the car isn't seeing the 'door open'. That will point you either in the direction of the door switch or the multi-plug connector on the door. I don't know about other models but my Abarth also has the "door open" warning on the digi display on the instrument cluster with ignition on - you can look to see if that works too?
I know my multi plugs on my drivers side door is an utter pain in the ass and keeps dropping temp sensor and window switches. Usually if I unplug and plug it in a couple of times it cleans the contacts up enough to work again for a bit longer. A good indication that it's messed up on my car is the outside air temp doesn't display.
I can't recall where the door switch is on the Stilo and the wife's just gone out in it so I can't check, but that sounds like a plausible first look. I've had one fail before on a Mazda and it kept setting the car alarm off in the middle of the night when it decided to think the door was opened.
If the switch doesn't prove anything you might have to plug it in and scan it to see if the body computer or door modules are flagging anything on the CAN network.
