Rulebook: What happens in ties during combat?
the player who has the lesser value decides if they want to add one or more bombs
First, this also obviously means "shields" as well; basically discard cards to get additional dice to roll. Several times in our game we ended up with tied rolls, I took the rule to mean we stop there as there is no "lesser value." Is this correct? Or is it meant as the "currently losing player?"
Bombs/shields are the same, and often are called bombs in certain situations (see also: storage bombs) and are named only dependent on whether you are attacking or defending (boms or shields). The rules need an update regarding ties, to correctly answer your question. The intented rules are written below.
The amended rules text should read:
the player who has the lesser value or unsuccessfully tied (since tie goes to the defender), decides if they want to add one or more bombs
Therefore, the attacker has the option of throwing additional bombs and continuing the attack, if the combat values were a tie. **These changes in wording will be added to the Rulebook 1.3 version to further clarify. Thanks to Chris Laudermilk for the question**