Is the problem that you can hear the shows? Offer a compromise for her to wear earbuds. Is the problem that she is watching these shows at all?

Usually someone who has obsessive behavior has something in their life that they're not settled with. The brain sort of "gets stuck" trying to redress the imbalance of the unmet need; the coping behavior (in this case the trashy shows) doesn't actually fulfill the need, but the brain's still latched onto it as what might help if you just get enough of it, so the consumption grows out of all proportion.

What's making her unsettled could be anything; could be work, or something in your relationship, or stress about money that's making it so she can't get settled mentally, or what have you. No way to tell from the outside. She may not even be aware of it. But I would try to get to the bottom of what it is and make it a priority. Trying to address watching the shows without addressing the underlying issue is going to be largely fruitless I think. Conversely, my guess is that once the underlying issue is addressed to some extent, then she won't be walking around with the phone or similar. She might still like the shows but it won't be to this obsessive level.

Plus then your wife will be happier, which is no small benefit.

Edit: So to address the I-think-pretty-reasonable question of how actually obsessive her watching of these shows actually is: How much does your wife do this? Like how many hours a day is she watching her shows? Does it actually interfere with other activities, or is it only something she's doing during "downtime" or chores or etc?

Just ask her to wear earbuds?

I don't think it's unreasonable to say that this kinda media really gets under your skin and ask if she'd be okay wearing headphones while watching it.

I have a similar relationship with romance reality and I've made it clear that I won't hang out on the couch if that's what's gonna be on TV. I tell my partner he's welcome to watch it all he likes, I'll just do something else and that's okay. He doesn't like the Gundam or mecha shows I watch, so I can just say "hey man, enjoy your show, but this is 'Gundam' to me." And that works as great shorthand. Sometimes he'll say "oh I was more interested in cuddling with you on the couch than really watching something, I'll change it" or "oh okay, I really wanted to catch up on this so you go have fun" and I'll go cook or play a game or workout.

Maybe there's a similar type of media preference you can reference to help your wife understand your actions.

Though while, ultimately, I think the onus lies on you to remove yourself from the situation rather than encourage your wife to change her behavior, it does sound like you're trying to do that with your own headphones. At some point, it might be on you to get better at filtering the noise, but your wife might be willing to help you by wearing headphones. That doesn't seem like a huge imposition to me, especially if it's for something I'm listening to while doing chores or something. For her listening to clips of shows in between activities? Eh, I think your part of the headphone compromise would be grinning and bearing those brief moments.

Lastly, while I don't think it'll directly address this issue, it's probably not a bad idea to do some introspection as to why this kind of media affects you so strongly. You clearly know what about it bothers you, but I'm sure you can think of other media types you dislike for a similar "moral disagreement" (for lack of a better phrase) that don't elicit such a reaction. It's nice to understand the things that trigger us in this way and why. Sometimes unpacking that makes them easier to bear and filter out, too.