Credit goes to Reddit poster at this link:
https://m.reddit.com/r/RemixOS/comments/48puc4/gapps_are_preinstalled_but_hidden_heres_how_to/
I have found that these Google apps are pre-installed on Remix OS:
Google Play Store - com.android.vending
Google Contacts Sync - com.google.android.syncadapters.contacts
Google Play services - com.google.android.gms
Google Services Framework - com.google.android.gsf
Google Partner Setup - com.google.android.partnersetup
Market Feedback Agent - com.google.android.feedback
Google Calendar Sync - com.google.android.syncadapters.calendar
Google Account Manager - com.google.android.gsf.login
Google Backup Transport - com.google.android.backuptransport
I also found a way to enable these, so you don't have to use something like GMSInstaller/GMSActivator, which could potentially be unsafe. If you did use one of these but want to use the stock apps instead, you will have to reinstall Remix OS.
This went unnoticed because there is an app called "JideAppPolicy" that disables them at startup. From what I can tell, this is the only thing it does. Also, apps disabled in a specific way do not appear in Settings.
Enter the root terminal (Alt+F1) and run this:
pm disable com.jide.apppolicy
This will disable JideAppPolicy. Then, run this:
pm list packages -d
This will list all disabled applications. To enable the Google apps, use this command:
pm enable <package name>
(Use the package names from the second command.)
Return to the GUI with Alt+F7. Don't open the Play Store yet. Check your date/time settings. Then, force stop and clear data for Google Play services and Google Services Framework. Restart Remix OS, and you should be able to download apps.
I'm not sure if Jide will release an update that removes Google apps, but if you want to change automatic update settings, check the Update > Options section in Settings.
This is working in the latest (beta) release... it might also work in the previous (alpha) release.
EDIT: Removed unnecessary step.