What probably is not that obvious is that the trophy does not imply leaving the stereo turned on from the moment you do it first to reaching yesterday. The description says "Reach yesterday without turning the stereo off", so you have to do just that, really. Meaning, you can turn it on and off as many times as you like, but it had to be on just before doing a certain action.
So, you can turn the stereo on (to proceed with the story), then turn it off immediately and still get the achievement, but only if you
turn it back on just before the end of the game. I also used Bernard to do his part the very last - after Laverne and Hoagie activated their time machines already.
Here is what I did in more detail:
*** Spoiler - click to reveal ***- Stereo was off as I already turned it on to solve a puzzle and then turned it off not to annoy me
- Activated the time machine for Laverne
- Activated the time machine for Hoagie
- Just before making a call to buy a diamond (yes, as late as that!), I went to the tentacles' room and turned the stereo on again
- Made the call to purchase the diamond
- Go the achievement
I just verified it the second time (first on Steam then on PS Vita). I think the important bit is that Bernard has to be the last character to do his part, as the very first time I played the game I didn't get this achievement, even though the stereo was on. But I did get it on both platforms (PC and PS) using the way described above.