Look at the Ketubah first. If handwritten all in Hebrew script you just sign in Hebrew Ploni ben Ploni, BUT it a printed commercial Ketubah that is in both Hebrew Script and English, you may be signing twice, once in Hebrew as Ploni ben Ploni, and again in English as John Doe.
My parents were wed in NYC more than 80 years ago. When mom went to file for Social Security benefits she was asked for a marriage license that was recorded with the city. Of course they didn’t have a marriage license, but they had a Hebrew/English Ketubah. The Social Security administration office made a photocopy of this witnessed (in English) document and awarded the benefits. B”H she collected for more than 30 years before passing away.