Given positive integers A_{1}, …, A_{t} and b ≥ 2, we write \overline{A_{1} ⋯ A_{t}}_{(b)} for the integer whose base-b representation is the concatenation of the base-b representations of A_{1}, …, A_{t}. In this paper, we prove that if (u_{n})_{n≥0} is a binary recurrent sequence of integers satisfying some mild hypotheses, then for every fixed integer t ≥ 1, there are at most finitely many nonnegative integers n_{1}, …, n_{t} such that \overline{|u_{n_{1}}| ⋯ |u_{n_{t}}|}_{(b)} is a member of the sequence (|u_{n}|)_{n≥0}. In particular, we compute all such instances in the special case that b = 10, t = 2, and u_{n} = F_{n} is the sequence of Fibonacci numbers.