Debugging asynchronous data fetching in Vue.js can be tricky due to the nature of asynchronous programming, where multiple tasks are executed concurrently. However, with some debugging techniques and strategies, you can identify and fix any issues that arise when fetching data from your API in a Vue.js application. Here are some steps to follow:

1. Verify the API endpoint: Start by checking if the API endpoint is returning the expected data. You can do this by using an API testing tool like Postman or curl, or by looking at the network traffic in your web browser's developer tools. If the endpoint is not working as expected, then there might be a problem with the API itself, which you should fix before proceeding with further debugging.

2. Check Vuex state management: In Vue.js, data is usually stored in the Vuex store for better organization and reusability. If the fetching of data is not happening as expected, it may be that the data is being accessed from a different part of your application or that there are some errors in the code. You can check if the data is being mutated correctly and if any errors are occurring when accessing it.

3. Use Vue Devtools: The Vue Devtools extension for Chrome and Firefox provides detailed information about your Vue.js app, including a timeline of events that occur during rendering and state changes. Using these tools can help you identify any issues with asynchronous data fetching in real-time.

4. Debug the code: If none of the above techniques work, you can try debugging the code manually by adding console logs to various parts of your Vue.js app. This will allow you to see what is happening at each step and identify any errors or issues that may be causing the delay in data fetching.

5. Optimize API calls: Sometimes, excessive API calls can slow down a web application. Make sure that you are not making multiple unnecessary requests, and consider batching your requests if necessary.

6. Test in different environments: Asynchronous data fetching can behave differently depending on the environment where it is being executed. Try testing your Vue.js app in different environments to ensure that it is working as expected in all cases.

Remember, debugging asynchronous code can be a challenging task, but by using the right techniques and strategies, you can identify and fix any issues that arise when fetching data from your API in a Vue.js application.