Can a player cast spells after losing priority but before the active player untaps?

Priority is the mechanism that lets a player cast spells or activate abilities. Under CR 117.1, a player may cast a spell or activate an ability only when they have priority. Once a player passes priority (or it is taken from them), they no longer hold it and cannot act until it is returned to them.

More specifically, the Untap Step (the very first step of a turn) never grants priority to any player. CR 502.3 states that no player receives priority during the untap step — permanents simply untap as required, and the turn moves directly to the Upkeep Step, where the active player then receives priority for the first time that turn (CR 503.1).

So there are two separate barriers here: (1) a player who has lost priority simply cannot cast spells or activate abilities until priority is returned to them (CR 117.3), and (2) even if we imagined priority were somehow relevant during untap, CR 502.3 explicitly forbids anyone from receiving it there.

Concrete example: It is your opponent's turn. During their End Step you cast an instant, then pass priority. Your opponent also passes, the spell resolves, and you both pass priority again, ending the End Step. The Cleanup Step occurs, then a new turn begins. Your opponent's Untap Step starts — no one gets priority, and you cannot cast anything. The earliest you could cast a spell is when your opponent receives priority in the Upkeep Step and then passes it to you.

In short, losing priority means you must wait for it to be returned, and the Untap Step ensures nobody acts there regardless — making casting during that window doubly impossible.