Honor system often compete, and since negatives out weight positives this usually means a downward spiral. However, a character can always strongly identify with one or a group of institutions or cultures. A cleric of peace for example, can take vows of non-violence, and thus avoid the penalties for turning down a duel that is insisted by his family.
One can declare an effort to focus on a group and after gaining 10 honour of rewards within the groups sub set one is now largely part of that group and other groups honour, both benefits and penalties are halved. Repeating this process for 10 net honour (not just honour this time) causes only your root culture honour awards to effect you, still at half.
Alternately you can gain at least 1 honour point and gain 3 levels while promoting the interests of the group you wish to measured by.
Note, more instituted groups can always cast you out of their system, thought this does not stop you from honoring their rules. The character Worf is one Star Trek example of this: conflicted with Star Trek vs. Klingon honour he was finally forced out a the Klingon system.