In most anime, enemies stay enemies. After all, anime villains are capable of some amazingly terrible things, and it would look pretty questionable if the heroes let them off the hook. However, occasionally characters who initially despised each other move past their feelings of hate, and start trying to form new loving connections. Whether the spat was caused by a mutual feeling of annoyance, or by straight up death match, romances that grow out of hatred can be surprisingly heartwarming.
When Shouko Nishimiya and Shouya Ishida of A Silent Voice were in elementary school, Shouya bullied Shouko relentlessly because she was deaf. Though Shouko tried to build a friendly relationship, her efforts failed, and the two parted as enemies. When they meet again as teenagers, Shouya feels terrible about how he treated Shouko.
Slowly, the two begin to develop a more positive, romantic relationship with no bullying involved.
For many Dragon Ball Z fans, Vegeta and Bulma seem like a supremely unlikely pair. The alien Vegeta starts off the series hating humans, and wants nothing more than to defeat his fellow Saiyan Goku. After being forced to work with Goku and his friends to defeat Frieza, his personality begins to soften, and he starts to develop relationships with humans, including Bulma.
Bulma isn't taking any of Vegeta's sh*t, but once he stops being so villainous, she's able to see the good in him. The two eventually get married and have two children, Bra and Trunks.
Kaname Ohgi and Villetta Nu start off Code Geass as mortal enemies. The fictional nation of Britannia has taken over Japan, and Villetta is a loyal Brittainian who sees the Japanese (called Elevens) as scum. Meanwhile, Ohgi is part of a Japanese terror group that is trying to take the country back.
Despite their vastly different political stances, and Villetta's unapologetic bigotry, the two eventually begin to recognize their shared humanity and end up falling in love.
Yuzuru Otonashi & Kanade Tachibana of Angel Beats are enemies largely because they don't understand each other. When Otonashi wakes up in the afterlife, he's told Kanade is a "destroying angel" who forces people to leave the afterlife before they're ready. As he gets to know her, he realizes she isn't a monster. She just oversees the inevitable processes of the afterlife, as everyone eventually must move on.
It turns out that the two are deeply connected. When Otonashi died, Kanade received his heart during a transplant, allowing her to live slightly longer than she might have otherwise. As these facts come to light, the two grow closer, and begin to develop romantic feelings for each other.