Okay, this is purely for me because I can't figure out Mukuro in the slightest bit. Hell if I know if I'm doing a good job or a bad job or if I should just stop. Mukuro is not easy :< Why did I do this to myself?
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So let's go into detail about what I think about Mukuro and how he interacts with others... and character by character because that's how my mind works. I'll kick this off by doing canon characters that are in the RP.
First we have the Yamamoto, who's a guy that I don't believe Mukuro has really talked to directly. I mean he's watched him from a far and heard things about him, but actually watching? Don't recall it. However, he is still a guardian and probably sees some of the world via Chrome but that still doesn't say much about how he'd behave. Though as a general rule of thumb for me, I believe that Mukuro would treat the other guardians a bit more... civilly than anyone he would encounter. I mean it's not like he'll be nice but he won't be as sardonic? ...well maybe not quite sardonic, but I'm using it for now.
Then we have Ryohei who I can honestly say that Mukuro kinda ignores. He may be a guardian and a memeber of Tsuna's family, but there hasn't been much that draws Mukuro to notice him. He wasn't one of the people that came with Tsuna to Kokuyo Land... and quite frankly, Mukuro doesn't really bother himself with anyone than those that could potentially help him in 'beautifying' the world. So being someone that doesn't show up on Mukuro's radar, he kinda just makes idle conversation with him. Rule of thumb still applies to him but Lussuria can have him and do all sort of questionable things to him... Which now puts the strangest images in my head, but I digress.
Though what I really should and want to talk about is his relationship with Tsuna and Chrome. Even if the RP doesn't have EITHER, I feel the urge to talk about it... Chrome to Mukuro is someone that he needs. Canonly. Without Chrome he's stuck in that god awful TUBE bound and chained for who knows how long. ...but thanks to her, he's able to not only exist outside of his confines but actually take part in something. I mean, he can do his walking around in dreams thing but uhh... as fun as that is, he's bound to get tired of it and want to actually DO something. So I'm assuming that Chrome to Mukuro is someone very precious to him. Yeah, kinda obvious, but I mean she's the only one that he has a fond nick/pet name for. "My little Chrome/Nagi". Even Ken and Chikusa, despite being a friends(?), are referred to as tools. Obviously he has a soft spot for the MOE.
Tsuna. ...After reading what Mukuro has to say about Tsuna I can't get over how WRONG and gay it is. "I will have his body"? "I will make him mine"? GAAAAAY. ...but uh, questionable words and homosexuality aside, I can't help but feel and believe that there's more to his desire to have Tsuna than what he says. I almost want to say that he wants Tsuna to take the ugly world and change it just as he changed him*. I won't say anything that'll make it concrete but I do believe that he isn't as "gun-ho" about "bathing the world in blood" as he was before. Won't say that he wants to be a part of Tsuna's family like everyone else, but he doesn't exactly want to be an enemy either. Corny line coming but, he's seen the awesome power of Tuna light? Tsuna is a mafia boss that doesn't want to BE a mob boss nor has any real ambitions. Yet when it comes to him, his friends and the people that love him and he loves, he doesn't sit idly by and let anything happen. As weak and pathetic as he is, he still tries to do something.
Mukuro never really saw anyone as a "friend" prior to his encounter with Tsuna. They're tools/servants, remember? But you really can't blame him for acting in this manner, his own family forced horrors upon him. Experimenting and eventually allowing him to remember all of his past lives... For him to see/hear a mafia boss to say such idealistic things, it had to have some impact especially after the purification.
Ex1. He allowed Ken and Chikusa to escape from the Vendicare at his own expense.
Ex2. Chrome. No really. I doubt that the old Mukuro would have saved her even if he sympathized with her.
So the way he treats Tsuna is definitely something different. He's kind in an unkind way and he respects Tsuna while also still believing that he's weakling that he is. As I've stated before, he sort of behaves as if he's a member of the family but at the same time not. It's almost as if he's in denial.
[*note: I believe that Tsuna purifying Mukuro's black aura had some effect on him. I can't say for certain what that is, but perhaps it allows him to not as cruel as he was before? There is a definite change in behavior from when they first meet at Kokuyo Land and when he comes to help Tsuna during the Mist Ring battle. Tsuna even recognizes that Mukuro is lying and has no ill intent when he says that he only helped to be able to get closer to Tsuna's body.]
.... and I think my brain is about to implode on itself. So I'll continue on this another time while also reviving this over 9000 times.