The creature in question eats foods rich in natural red dyes
Most of these plants require boiling to get the dye, but the acid bath of digestion is enough to get to that point. The problem, of course, is that the mammal would need to eat this stuff almost exclusively (or at least a whole lot of it). But methinks it would create the color you're looking for.
- Autumn red leaves (seasonal)
- Bedstraw Root
- Beet Root
- Canadian Hemlock
- Choke Cherries
- Crabapple Bark
- Hibiscus Flowers
- Madder Root
- Rose Hips
- St. John's Wort (a bit harder, requires mixing with isopropyl alcohol)
- Sumac Fruit
- Sycamore Bark
(Found from this source)
But what about the thicker fluid?
I believe you'll need to live without this. Mammary glands are designed to create and pass a relatively specific fluid thickness (not unlike the kidneys or liver). If you try to force something substantially thicker through the glands, they get all clogged up (not unlike the kidneys or liver). Getting something the thickness of blood that didn't hurt the lady would require a physiological change to the glands.
You could thicken the fluid somewhat by consuming high-starch foods, but I believe that's the best you'd get.
1 "It is by my will alone that I set my mind in motion...." Sorry, I had to include a gratuitous "Dune" reference. Eating enough elderberries to turn milk red might turn the whole blooming cow red.