Thanks very much for the responses.
I really don't ever give in as far as giving her something she's screaming for, but I do sometimes lose it and give her a negative reaction like yelling. So that's probably working out as a payoff for her of some kind.
She truly hates being alone so putting her in bed should work well during a tantrum. And I really like the role-playing idea- she'll love that. Also very good to know that many of you have experienced this and come out in one piece on the other side. Thanks!