Is it possible to run more than one site on the same WordPress core and a set of plugins?
I want to be able to have custom plugins and themes for each site, but have a base set that is the same, to make it easier to maintain.
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Yes, it is possible.
You can setup subdomains sites like this:
Then, after you setup the network, use a domain mapping plugin to setup a domain name for each of them.
There are several domain mapping plugins out there:
Maybe this WordPress MU Domain Mapping (Wordpress Plugin) is handy for that? Just like running a Multisite installation and offer Plugins in the Network.