Modern energy infrastructure has evolved into an integrated electricity and natural gas systems (IEGS), which often encompasses multiple geographically-diverse energy areas. This paper focuses on the decentralized adjustable robust operation problem for multi-area IEGS. Existing distributed algorithms usually require synchronization of all area subproblems, which is hard to scale and could result in under-utilization of computation resources due to the heterogeneity of local areas. To address those limitations, this paper proposes an asynchronous alternating direction method of multipliers (ADMM) based decentralized model for multi-area IEGS. This asynchronous decentralized structure only requires local communications and allows each area to perform local updates with information from a subset of but not all neighbors, where the individual areas' subproblems are solved independently and asynchronously. Meanwhile, the linear decision rules (LDRs)-based adjustable robust operation model is tailored to combine with the automatic generation control (AGC) systems to fully exploit its potential in dealing with renewable energy uncertainty. Numerical results illustrate the effectiveness of the proposed method.