Humility is a critical component of our Catholic spiritual life. The goal is to be a meek servant of God and not draw attention to yourself or glorify your sentiments (e.g. "self-expression"). In that way, tattoos are like a venial sin.
Leviticus 19:29 states, Do not cut your bodies for the dead or put tattoo marks on yourselves. I am the LORD
That is pretty clear.
1: Corinthians 6:19 states "know ye not that your body is the temple of the Holy Ghost which is in you, which ye have of God, and ye are not your own?"
I take that to mean our bodies belong to God. I don't hink God appreciates people tagging up his house without permission. That said, for me personally if she has tattoos already before she was aware, that would probably be fine (although I do find them distasteful). It would be a deal-breaker for me if she saw nothing wrong with continuing to get tattoos.