Select Checkout
Magento should take to https checkout page.
Takes the user to the http account page.
Did you make sure you have Allow Guest Checkout set to "Yes" under Store > Configuration > Sales > Checkout > Checkout Options?
@OgreMHDW Thank you for reporting. Please let us know how did you configure checkout.
"Allow Guest Checkout" is not enabled. "Require Customer To Be Logged In To Checkout" is enabled.
So I think I may have figured something out. This appears to be an issue with the "Enable Onepage Checkout" option. I have set onepage checkout to No and now am able to view and edit cart without it being emptied. Unfortunately, I have no way to check out at that point.
Here is the view cart setting with Enable Onepage Check enabled:

Once I select Go To Checkout, it takes me to the account page with cart cleared.
Here is it disabled:

I have no way of checking out at this point.
So an update. I figured this may have happened due to installing from Composer. I created a staging site and used the downloaded Magento 2.0.5 for installation. The site was working normally until I enabled SSL. After that I am experiencing the same issues as my other installation. Since SSL is critical to protecting passwords, this is not functional without it. No caching is being used on the new staging install.
This is also happening on 2.0.6.
@OgreMHDW Unfortunately, I could not reproduce the issue as you described it. May you please advise if this problem still relevant for latest version of Magento2?