Opponent cast Beseech the Mirror using Bargain. They searched for and cast Day of Judgement. The game did not give me priority to respond to Day of Judgement though I had Earth Kingdom Protectors with numerous other allies in play, and untapped lands that could cast an instant in my hand.
I assume the problem here is the nature of casting as a part of resolving Beseech. The second spell still goes on the stack so there should be time for responses.