Honestly I don't know that there is a Rem 870 in the double barrel world.
Most people will be quick to recommend either a Browning or Beretta, but you're going to be looking at an entry price of around $2,000 for a field model. My dad has had great luck with his 12 and 20 Silver Pigeons over the last 10-15 years and I've been happy with my 20 Citori for about the past 3. I've put thousands of rounds through the Browning and each of the Berettas have to have tens of thousands on them.
I personally won't recommend a Stoeger if you're looking for a quality long term gun; both my dad and a friend of mine had a firing pin stop working. My friend's was out of warranty and they didn't even bother having it looked at due to the initial price of the gun. My dad's may have actually been in warranty and was either fixed and failed again and/or wasn't worth the hassle of dealing with the warranty. If you want something cheap and don't care too much about it lasting though, it might not be a bad choice.
When I was shopping for my Citori I also considered the Franchi and CZ guns pretty heavily. I've read decent things about them, but didn't know anyone with any experience with them. Another friend of mine does have a Turkish .410 but as Robert mentioned that distributer stopped importing years ago, nice gun though wish you could still get them. I believe CZ is actually still sourcing their guns out of Turkey for what it's worth.
At the end of the day I wanted a gun that I knew would last and ended up with the Browning since I shoot left handed and the Berettas are cast for a right hander. In short Beretta or Browning if you can stomach it, maybe take a look at the CZ or Franchi for something a bit cheaper, but I'd stay away from the Stoegers.
Edit: You could also look at the used market to bring the price down on a Browning or Beretta. I had a really tough time finding much of anything in 20 ga though, most everything out there I found was in 12 ga.