Other hunters treat their hybrid partners more like tools, or dangerous beasts. He tries to give her as 'normal' a life as possible for the time she has. The main character, a young man, is keeping it a secret from his hybrid partner how close she is to transforming, to keep from upsetting her. If I recall correctly, the hybrids have an animal 'theme' - the main character relies on speed and so is rabbit themed, another relies on stealth and is mantis themed, and so on. The hybrids have superhuman strength and monster-related powers, but every time they use their powers it brings them closer and closer to a tipping point where their monster genes will take over, and they'll transform into full monsters. The hybrids are all females, and all are little girls (as the monster invasion only happened within the last ten years). The hunters are teams of two: one adult human and one half-human, a hybrid of human and the monsters.

If a monster does manage to get past the towers - which is happening more and more often as the story starts - there are dedicated monster hunters to combat them. The main city, possibly used to be Tokyo or another major Japanese city, is protected by a ring of huge towers, the material of which repels the monsters.

The premise is that the world has been overrun by nightmarish monsters - huge, dark, very Lovecraftian with tentacles and insect-like parts, and glowing red eyes. The version I saw was dubbed into English. I'm fairly sure this anime came out within the last 5 or 6 years.

Anime where legends fight monsters