A man may give whatever he wants of his wealth to his wife, and the wife may give whatever she wants of her wealth to her husband, as Allaah says (interpretation of the meaning):
“And give to the women (whom you marry) their Mahr (obligatory bridal-money given by the husband to his wife at the time of marriage) with a good heart; but if they, of their own good pleasure, remit any part of it to you, take it, and enjoy it without fear of any harm (as Allaah has made it lawful)”
If you willingly give your half to your wife, and register the building in her name, then she will be the sole owner of it.
If she willingly gives you her share, then register the building in your name.
But if each of you wants to keep his or her own part, then register it in both your names.
The point is that the registration should reflect the real situation. That is what is best, so as to protect your rights and avoid conflict.
If the building has to be registered in the name of one person only, then the other partner may protect his or her rights by having an official document that states that and is signed by witnesses.
May Allaah help us all to do that which He loves and which pleases Him.
And Allaah knows best.