I would argue that there's a huge overhead. At least if you make use of tools that these companies provide. And if you don't, then just go with dedicated boxes which you can get cheaper than cloud solutions.
I think this article[0] is very well written and clearly explains the problems with provisioning resources in multiple clouds. Yes, there are tools that allow you to do it, however you give up most of the reasons for going into the cloud.
Switching between cloud providers is much easier than going from on-prem to the cloud. So, once your systems are "cloud" ready you should be able to avoid vendor lock in naturally.
