Asynchronous picture and sound means that the video and audio portions of your AVCHD file become increasingly out of sync as the video plays.

1. Start by copying (not dragging) the related AVCHD files (e.g. 00653.MTS, 00654.MTS, etc.) to a new folder. 

2. Download Total Commander. This program costs money, but a trial version is available which you can try for as long as you like.

https://www.ghisler.com/index.htm

3. Install the program and open it.
On the left, navigate to the newly created folder.
Select the AVCHD files.
On the right, select the folder you want to use as your destination folder.
Select "Files" > "Combine Files". A new file will appear in the destination folder; it will not have a file extension.
Rename the file and add the extension .MTS to the end of the name.

4. Import the newly created file into Movie Edit Pro or Video Pro X and play it. The picture and sound should now be in sync. 

Be sure to only combine related videos.