Here's the deal ... Rihanna was photographed this week in a full-length fur coat, which PETA tells us is almost certainly made of the real deal ... and now the animal rights org wants RiRi to make some fashion changes.

PETA fired off a letter, obtained by TMZ, to Rihanna's team ... saying they noticed her fur coat, then pled their case for switching to faux fur and offered up a faux fur coat.

In the letter, PETA's senior vice president Lisa Lange cites Rihanna's growing family, writing ... "As a mother, you know what it's like to feel protective and to want to make sure no harm comes to your family. Please understand that this desire -- this instinct even -- is shared by the minks, foxes, and rabbits who are torn away from their families and killed for their fur."

PETA then asks Rihanna, "Won't you please help create a kinder world for your children to grow up in by deciding today that you'll no longer wear fur?"

The org offers to take any of Rihanna's fur clothes off her hands and donate them to survivors of the recent earthquakes in Syria and Turkey ... PETA says those folks truly have an excuse to wear real fur because they lost everything and now have to survive winter.

PETA also tells Rihanna they're sending her a faux fur coat from Unreal Fur, which they say will help her "stay warm and fashionable while making the cruelty-free switch."