In one of the most helpful summaries of Christian engagement and navigation of the Graeco-Roman world, Barrett outlines the specifics of such a negotiation and how the Christians were to be loyal to Christ and yet not abstain from interactions with those outside the Christian family.
Paul did not ask his converts to come out of the world; he did not even ask them to abstain from non-Christian dinner parties, though he was aware that these could constitute a problem. He did not expect marriages to be broken up on the ground that only one of the partners had become a Christian; Christian and non-Christian (unless the latter took the initiative) should continue to live together. A widow, remarrying, should, however, exercise a Christian choice. Paul could, in the interests of the Gospel, live like a Gentile, and it was possible for unbelievers to find their way into the Christian assembly. On the other hand, Paul warned his readers against the practice of taking part in meals in idol-shrines, and expected them to settle their own disputes without making use of non-Christian courts; and one must remember the moral break made by conversion (1 Cor. vi. 9 ff.), and the separate existence of the church as the community of God's elect (1 Cor. i. 1-9). The position was anything but simple. The Christian was in the world, but must remember that the outward shape of this world is passing away (1 Cor. vii. 29 ff.). He could not but live in the midst of unbelievers, and must live in contact with them since in this way he might hope to save them (see 1 Cor. vii. 16 for a special case); but he himself was a member of the holy people, who would judge the world (1 Cor. vi. 2 f.).
 C.K. Barrett, The Second Epistle to the Corinthians. (London: A. & C. Black, 1973), 196.