Interval fuzzy preference relations (IFPRs) that can simply denote the lower and upper bounds of decision makers' uncertain judgments are keenly studied by researchers and are widely used in practical decision-making problems. Consistency analysis is vitally important to avoid illogical ranking orders. First, this paper analyzes issues in previous additive consistency concepts for IFPRs. Then, it presents a new additive consistency definition that overcomes issues in previous ones. A linear programming model to judge the additive consistency of IFPRs is constructed, and a method to derive additive consistent IFPRs is proposed. Furthermore, a goal programming model to determine missing values in an incomplete IFPR is built, which have the highest consistent level with respect to known values. Regarding group decision making, a group consensus index is defined to measure the consensus of individual IFPRs, and an approach to improve the consensus level is introduced. Finally, a method for group decision making with IFPRs is developed, which can address incomplete and inconsistent cases. Associated examples are given to show the efficiency and feasibility of the developed theoretical results and comparative analysis is made.