With this extension, from the Microsoft DevLabs team via the Marketplace, you can resolve these conflicts online, as part of the pull request process, instead of being forced to break flow and resolve locally.
After adding the extension the new conflicts tab is visible which enables conflict resolution in the familiar side by side review page as shown below:
Really nice extension, which should make resolving merge conflicts a much more straightforward part of the DevOps workflow.
Additional - Adding the extension from the Marketplace
For those that have not used the Marketplace for Azure DevOps as yet, below is a brief tour of the process.
Note* For a seamless experience, login to Azure Devops first.
1 - Navigate to the Marketplace and select the extension you would like to add.
2 - Select your DevOps Organisation
3 - Navigate to your DevOps project
4 - Navigate to the pull request that contains the merge conflict
5 - Select the conflicts tab as shown above.