Tight line, rod pointed straight at the flies and a pretty fast retrieve. A big rainbow came and ate both streamers before I could strip set on the take. First fly was in its throat.
Barbless hooks make this not nearly such a big deal though. With hemostats and little skill Barbless flies can be removed when taken deep with very little additional harm if any.
Anecdotally, I once caught a big tiger trout that had thick mono going down its throat. It had clearly swallowed a baitfisher’s hook and I wasn’t going to be able to get it out. But then I noticed it had pooped out a big bait holder hook (the kind with reverse barbs on the shenk to hold worms on) that was still connected to the hook. I cut off the hook and the line slid right through. If he could pass that big hook, they can do more than I thought in getting hooks out. It was only found by me because it was connected to 18” of line.