I had never thought about this before, but when Paul mentioned himself and Barnabas:

That this was a good while after he and Barnabas had previously parted.

Apparently the Corinthians knew of Barnabas (maybe he had come through there?)

Also, it seems that Paul was speaking of Barnabas in a good light, about him and Paul in the same sentence.

So that, perhaps in the minds of the Corinthians, they would always think of both men as being of one accord, rather than opposed?

Also, is it possible that the events of:

Galatians 2:11 But when Peter was come to Antioch, I withstood him to the face, because he was to be blamed.

  12 For before that certain came from James, he did eat with the Gentiles: but when they were come, he withdrew and separated himself, fearing them which were of the circumcision.

  13 And the other Jews dissembled likewise with him; insomuch that Barnabas also was carried away with their dissimulation.

  14 But when I saw that they walked not uprightly according to the truth of the gospel, I said unto Peter before them all, If thou, being a Jew, livest after the manner of Gentiles, and not as do the Jews, why compellest thou the Gentiles to live as do the Jews?

occurred in the time between:

Acts 15:35 Paul also and Barnabas continued in Antioch, teaching and preaching the word of the Lord, with many others also.

and

Acts 15:36 And some days after Paul said unto Barnabas, Let us go again and visit our brethren in every city where we have preached the word of the Lord, and see how they do.

If so, then perhaps the tension and rebuke described in Galatians, was still fresh, when the events of:

Acts 15:37 And Barnabas determined to take with them John, whose surname was Mark.

  38 But Paul thought not good to take him with them, who departed from them from Pamphylia, and went not with them to the work.

  39 And the contention was so sharp between them, that they departed asunder one from the other: and so Barnabas took Mark, and sailed unto Cyprus;

  40 And Paul chose Silas, and departed, being recommended by the brethren unto the grace of God.

occurred?

.