Either the planet or the previous faction should get the shipyard. Maybe the player can pick which planet gets the shipyard or otherwise I agree delete the thing. 

I'd like it if the AI would decommission shipyards no longer in it's territory and I think we shouldn't even see the rally points of other players.

Shouldn't there be some 'sneaky' backstabbing diplomatic missions be availible to avoid a planet to switch sides?

I would prefer getting it...but I would like them gone, even if I've never culture flipped anything but asteroids. :3

I've invaded and/or flipped a bunch of worlds. The enemy rally points persist and I can't delete them. It looks silly.

I have been playing the culture flipping method a lot in the past two weeks and I totally feel the shipyard should be flipped along with the primary planet sponsor. It makes the most sense as it really is an extension of the planet.

Perhaps upon culture flipping the shipyard could be inactive for several turns until it is fully operational under the new owner.

But yes, I hate the Shipyards just sitting there in my territory - and I can't attack them without declaring war. At the VERY lease the AI should migrate them within their own borders. But that seems overly complicated so changing ownership still seems more logical.

If a shipyard is sponsored by multiple planets and you flip one of them, should the shipyard be flipped then?

@eviator - No, if it still has remaining sponsors it shouldn't flip or disappear.

Okay, about the rally points, don't you think it would be better to not show it at all for other civs? That takes care of that.

@eviator: this is a very interesting point. so far I know the rally points are just imaginary for empires internal use. It would also get an tactical part to find out where enemy empires have their rally points instead of just getting free intel at the map.