A Muslim woman is not allowed to marry a non-Muslim man whatever his religion is. There are clear and explicit Islamic injunctions on this point in both the Quran the sunnah. Allah the Almighty said: “…and do not marry (your girls) to idolaters until they believe…” (Al-Baqarah: 221) And He said concerning the immigrant Muslim women: “Then if you know them to be Believers, do not send them back to the unbelievers. They are not lawful for them (as wives), nor are they lawful for them (as husbands).” (Al-Mumtahanah: 10)
Hence, it is not enough for a non-Muslim to be a Muslim to just have inner faith in the Islamic beliefs. This unannounced belief may benefit this person in the Hereafter if he cannot announce his faith for fear of being killed or the like cases of necessity, but he still has to observe as much of the Islamic obligations as he can, such as observing salah, fasting in Ramadan etc.
Apart from the cases of necessity, the non-Muslim has to pronounce the shahadah and announce his conversion and his embracing Islam in order for the Muslim community to treat him as a Muslim and give him his rights. There are rights, duties and obligations follow the announcement of embracing Islam which cannot be fulfilled with keeping this as a secret.
For the validity of marrying a Muslim woman, the husband has to be known as a Muslim. This issue is not subject to compromise.