From a SeekersHub post Advice to a Christian Man Who Wants to Marry a Muslim Woman by Ustadha Zaynab Ansari, we have the following:
Given that you’ve clearly thought this situation through, I would now offer that the onus is on the woman whom you wish to marry. After all, your religious beliefs may not prevent you from marrying outside of your faith but hers do and it is her responsibility to consider the implications — whether spiritual, social, or legal (were she to travel to a Muslim-majority country) — of marrying outside of her faith.
I'm particularly interested in the legal implications:
Question: What are the legal implications of a Muslim woman marrying outside her faith?
The fatwa does not go into detail. I'm not clear on what the fatwa is talking about, since presumably non-Muslim marriages are recognized as marriages in Muslim-majority countries. Nor am I aware of any sharia punishment for such a marriage.