Captain Kilowatt is correct - since your fundamental signal is causing the problem, no amount of filtering on your end - save turning your radio off - will fix this. You have to eliminate your signal at the neighbor's phone.
Here's what's happening: the wires act as antennas and pick up your 27MHz signal that travels into the electronic appliance, where it encounters the p/n junction of a solid state device. That p/n junction acts like the detector diode in a radio, turning your RF into an audio signal that's amplified right along with the voice/music/sound the electronics were made to reproduce. And that's the big problem, because once it's rectified into audio you can't do anything about it.
It doesn't help that the biggest complainers are usually also the penny pinchers who buy the cheapest crap they can find. And one way manufacturers save cost on bottom of the line junk is by eliminating RF filtering.
Back in the olden days I used to buy .1uf ceramic caps by the bagful and stick them everywhere. Across phone lines, between the mouthpiece screw terminals of old Western Electric 500 phones, across stereo speaker terminals, etc. The whole idea is to give a short circuit to the RF that won't affect the audio frequencies.
I had neighbors who used to bitch about my little 2 watt Midland basically coming through on just about every piece of electrical gear they had. I put caps on all their phones, and 3 caps on their stereo speaker leads (one across the leads and one each to ground). I couldn't do anything about my signal overwhelming the front end of their P.O.S. AM radio the wife listened to in the kitchen as she drank her daily quart of vodka, and ended up just buying her a nicer model so she would stop beating on my door in a drunken rage every day.