Eulogy's prostitutes are two slaves caught a few months back. They have been systematically brainwashed and threatened until they've become completely subservient to their master. Eulogy verbally abuses them constantly. You can listen to the tirades as the girls meekly plead for forgiveness. When Eulogy is away, they're both catty to each other. You cannot speak to either of them; any conversations must be through Eulogy.
Crimson is one of these slaves. She serves as Eulogy's bodyguard, but she often talks back to him when Eulogy speaks down to her. She doesn't appear to like her fellow slave Clover, and the two often bicker.
If the player chooses to kill Eulogy Jones without recruiting Clover, they both may attack the player instead of escaping with their freedom.
Sometimes you'll find Crimson dead on the top, outer balcony of Eulogy's pad, killed in a duel by Clover; if you have high sneak and/or a Stealth Boy, it's sometimes possible to watch this duel happen.
If you cause the slavers to become hostile toward you outside of Eulogy's pad and then enter it, you may find Eulogy killing Crimson by shooting her in the head.
Crimson doesn't seem to be as successfully brainwashed as Clover is, and will often talk back to Eulogy.