The number of zeros and the number of ones in a binary string are referred to as the composition of the string, and the prefix-suffix compositions of a string are a multiset formed by the compositions of the prefixes and suffixes of all possible lengths of the string. In this work, we present binary codes of length n in which every codeword can be efficiently reconstructed from its erroneous prefix-suffix compositions with at most t composition errors. All our constructions have decoding complexity polynomial in n and the best of our constructions has constant rate and can correct $t = Θ(n)$ errors. As a comparison, no prior constructions can afford to efficiently correct $t = Θ(n)$ arbitrary composition errors. Additionally, we propose a method of encoding h arbitrary strings of the same length so that they can be reconstructed from the multiset union of their error-free prefix-suffix compositions, at the expense of h-fold coding overhead. In contrast, existing methods can only recover h distinct strings, albeit with code rate asymptotically equal to 1/h. Building on the top of the proposed method, we also present a coding scheme that enables efficient recovery of h strings from their erroneous prefix-suffix compositions with $t = Θ(n)$ errors.