Implied consent and the meaning to put upon "upon"

In Roell v. Withrow, dealing with the constitutional requirement of consent to entry of final judgment by a magistrate judge, the majority of the Supreme Court concluded that "the better rule is to accept implied consent where, as here, the litigant or counsel was made aware of the need for consent and the right to refuse it, and still voluntarily appeared to try the case before the Magistrate Judge."

The dissent says that the statutory language "[u]pon the consent . . ." means that the consent must come first.

I wonder if the justices would have been aligned differently had the issue been whether consent to a police search could be implied from the fact that the police said several times they thought they had consent and the citizen said nothing until after the search was over.