The rules say,
Number of objectives
This reward may be claimed by any player at the moment he controls the exact number of corresponding objectives
To complete an objective
An objective is completed as soon as all its spaces are occupied by a legion. The player then announces AVE CESAR.
Then he proceeds as follows:
1. He removes all the legions from the objective and puts them back into his stock.
2. He carries out the possible power provided by the objective.
3. He moves the objective into his area showing the objectives under control.
4. If possible, and if he wants to, he claims a reward.
The claiming rewards stage is clearly at the end. Is there a moment when he can claim the 5 bonus tile, before completing the 6th card, for example?