I realize that's a big ask (and is probably something the operating system can't do), but it doesn't violate any laws of physics.
I'm not really sure it's feasible, though. I don't know the details of how phones interact with towers, but I would think that what's happening is that your phone broadcasts a radio signal, and that same exact signal is picked up by more than one tower. If you want to triangulate the location of the radio signal, you don't need to know how long it took to get to the tower, you just need to know how much longer it took to get to one tower than the other, and the locations of the two towers.
Additionally, I think you can make a similar calculation based on signal strength if you can account for anisotropies in the attenuation of the signal. This seems less reliable to me, but if this is how it is done you could potentially throw off your location by a little bit by putting your phone in a partially shielded case that increases the attenuation on one side relative to another.
I don't think what you're suggesting would actually work given the way mobile network infrastructure works (where multiple cell base stations may see each message at the same time).