Honestly, friends little sisters are sooooorta off-limits but (legal) love is love so no judgement. She sounds pretty irrationally pissed off so I doubt you would get far trying to sit her down for coffee and apologizing but I would definitely try and apologize since she is upset and try to understand her feelings. Maybe trying to downgrade the hateful communications to a brief semi-rational conversation to at least understand WHY she is upset would be a good start. From there try to help her understand your reasoning and actions, she is going to be immensely protective of a little sister so you are starting with a huge disadvantage but do what damage control you can.

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If that doesn't work out, either way there are legal reprocussions for harassment and slander so that is an available if not undesirable outcome. I would try and be as polite and open as you can because if she doesn't cool off this still has a while to play out.