The hosts have been unable to dislodge the pairing, who have now built a partnership of 141 runs.
Compton (73) was the first man to reach his half-century, despite nearly cutting the ball onto his stumps when on 48.
However, he drove the ball down the ground for two off the next delivery.
Trott (68) brought up his half-ton in 106 balls.
Trent Boult has been the most expensive of the Black Caps attack, going for 49 in his 11 overs with just one maiden.
Meanwhile, Bruce Martin has gone for just 23 in his 16 overs with seven maidens.