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I personally think she is Anokata's sister or daughter, cannot be wife/lover due to her implied sexual relationship with Gin and knowing Anokata's over cautious nature and his favoritism of her would hint otherwise. She cannot be his mother or aunt or any older generation because the BO is over half a century and that would put her over 60 years of age, odd for a woman in that age to be active in 'such' relations. Therefore she is as close as a cousin, sister, or daughter. I'd say as daughter or sister he would be over protective of her, but then again he trusts Gin. My initial thought regarding Vermouth as daughter or sister is the fact that she is very very close to be under that rebellious daughter cliche, that would be the father's workaholic nature and specially in the underworld causing the 'death' of her mother. Anokata therefore would try to acquire as much money as he can by any means to support scientists in any field to help regain his wife/ Vermouth's mother, back to life, but his ignoring of his daughter's welfare caused her to rebel. Or the way he is becoming very obsessed with the idea of revival and longevity rather than focusing on the health of his dead/unconscious/decapitated/or something wife, so it is most probably because of his underworld work that his wife got caught in an attempt of murder leading to Vermouth's despise of the Org/Anokata. That is if this weak theory holds, however I cannot think of a daughter addressing her father as boss unless he is strictly formal and does so to hide their intimate relation. But how cool would it be if such is the case ..