In this paper, we focus on a bidimensional risk model with heavy-tailed claims and geometric Lévy price processes, in which the two claim-number processes generated by the two kinds of business are not necessary to be identical and can be arbitrarily dependent. In this model, the claim size vectors (X 1 , Y 1) , (X 2 , Y 2) , · · · are supposed to be independent and identically distributed random vectors, but for i ≥ 1, each pair (X i , Y i) follows the strongly asymptotic independence structure. Under the assumption that the claims have consistently varying tails, an asymptotic formula for the infinite-time ruin probability is established, which extends the existing results in the literature to some extent.