How do you combine qualitative and quantitative data when they disagree?

Treat them as answering different questions: analytics tells you what is happening and at what scale, qualitative work tells you why. When they disagree, first check the measurement (event definitions, segments, date ranges) because instrumentation is wrong more often than people expect. Then look for a segment effect, since an average can hide two groups behaving in opposite directions.

My first move is not to pick a winner, it is to check whether we are even measuring what we think we are. I have lost count of the times a scary drop turned out to be an event that stopped firing after a release, or a funnel that counted a step twice. So I go and read the event definitions with an analyst or an engineer before I argue with anyone. If the measurement holds, I segment. Averages lie constantly. On one project the completion rate looked flat after a change, and when we split by device it turned out desktop had gone up meaningfully and mobile had gone down by almost exactly as much. Those are two different stories and the average had erased both. The interviews then explained the mobile side: the primary action sat below the fold on smaller screens with the keyboard open. That is the pattern generally. The numbers tell me where to point the camera, the sessions tell me what is happening in the frame, and neither one alone was enough to make the call.