I have a australian red centre which is a cross between a finger lime (Citrus Australasica) and a Burmese Rangpur lime, ( Citrus x Limonia). I have seen it also marketed as blood lime and red lime.

I also have a red finger lime which is variety of Citrus Australasica.

As trees, they are similar. They have small green leaves and lots of thorns. I've had my finger lime for 6 years in a large pot. I keep it pruned because of the thorns but its relatively easy to grow

The finger lime is delicious and produces little red caviar like pearls of juice. The color outside is dark green but inside they range from pinkish to red. It produces tons of fruit.

The red centre lime is red on the outside (and in the inside?) But i have yet to try the fruit as my tree is little. I'll post pictures when it does eventually fruit.

There is a huge thread on this where the 3 different limes with a similar name, get hashed out and you will find folks that have at least 2 or all 3 - https://www.gardenweb.com/discussions/4949487/ordered-a-red-lime-tree-on-a-whim-anybody-growing-one?n=136

In summary -

The generic "red lime" appears to be this (or similar) - http://www.citrusvariety.ucr.edu/citrus/excalibur.html

The "blood/red centre lime" appears to be this - http://www.homecitrusgrowers.co.uk/australiannativecitrus/bloodlime.html

And Saltysea mentioned the finger lime, where no one can find the red varieties here in the U.S. (at least yet). Vendors are mainly selling the green one (with clear vesicles) - http://www.citrusvariety.ucr.edu/citrus/microcitrus_australasica_3672.html

As a note, I have the red centre one. It's still in the adorable stage but has a similar growth habit as the finger lime (with the tiny leaves and many long thorns). Got it from Logees.

Hmmm, I just ordered a red lime from Hirt's. It says it is a cross between a Rangpur lime and Kumquat, orange skin and reddish inside. About the size of a key lime.

Oh, I would love a red finger lime, but around here, (USA), you are lucky to find the green one. I did get some seed from a red one, (I hope), and got three to germinate, out of six. Now just have to wait the years, to see what I really have. Lol!

I have the red lime from Hirts, but no fruit yet. I also have the blood lime (or whatever they call it) that Logees sells. I have had fruit from that one. It has rounder pods than the finger lime and instead of caviar (round), they are elongated pods that push out. It looks a lot like the green finger lime I have. I will say that the flowers are lovely and have a wonderful rose-like scent. The fruit is pretty good - very strong. It would work well as a toping accent for fish or other citrus-infused meats or desserts.