Assuming all the installs are on the same instance and thus can have the same files shared between them, then the main thing you need to do is to have all the custom content stuff live outside of the main WordPress folders.
So first, you’re going to want a fresh copy of WordPress somewhere, untouched (and untouchable, the whole point is to have a single copy, right?).
Next, you’ll use hard-links or sym-links to mirror those files elsewhere. Normally in unix-like environments, you use ln -s to do symlinking. How to do this specifically is left to the reader, but essentially if you have your WordPress in /example/wp, you would do something like ln -s /example/wp /home/username/public_html/wp or something like that. If sym-links cause issues, switch to hard-links instead.
Now, the main two things that live inside the main WordPress folder, but which are custom and not part of “core” are your wp-config.php file, and the entire wp-content folder. So, we need to get these outside of the main /wp folder for each site. Both of these are possible.
The wp-config.php file is easy, because WordPress also checks a directory up from itself by default. If your wp folder is at /home/username/public_html/wp, then WP will check for /home/username/public_html/wp/wp-config.php, but failing that it will look for /home/username/public_html/wp-config.php as well. So just make your wp-config.php files in the directory above.
For the wp-content folder, you can reposition it outside the main wp directory by adding code to your wp-config.php file like the following:
define( 'WP_CONTENT_DIR', '/home/username/public_html/custom-content' );
define( 'WP_CONTENT_URL', 'http://example.com/custom-content');
And then WordPress will expect the content and everything in it to live there instead. The wp-content folder inside the /wp directory will be ignored completely, so you’ll have to make this folder and put your themes/plugins/uploads in there.
The last step is to use .htaccess (or some other config files if you’re not using Apache) see the /wp folder as visible from the root, as if it was there. This isn’t tricky, it just requires a bit of thought. See, the normal WordPress .htaccess rules (for non-default permalinks) essentially tell Apache to redirect all requests to the main WordPress index.php file if the request isn’t referencing a file that actually does exist. This allows uploaded pictures and such to work directly, outside of WP.
Here’s an example ruleset you could use in the /home/username/public_html/.htaccess to mimic this, but also account for the /wp directory.
Options -Indexes
RewriteEngine on
RewriteCond %{HTTP_HOST} ^(www.)?example.com$
RewriteCond %{REQUEST_URI} !^/wp/
RewriteCond %{REQUEST_FILENAME} !-f
RewriteCond %{REQUEST_FILENAME} !-d
RewriteRule ^(.*)$ /wp/$1
RewriteCond %{HTTP_HOST} ^(www.)?example.com$
RewriteRule ^(/)?$ wp/index.php [L]
Essentially, these rules say two things.
First: If the requested URL doesn’t have /wp/ in it and it’s not an existing file or directory, then change the URL using an internal rewrite to have /wp/ in front of it. This has the effect of remapping all URLs from http://example.com/whatever to http://example.com/wp/whatever. Basically, it makes it work as if /wp/ was actually in the root of the site.
The second rule changes requests for the root url (just the /) to the /wp/index.php file. This may not be strictly needed, but it works for me.
This allows the /wp directory to remain “pure”, as it were, and thus you can use symlinks or hardlinks to keep the one-true-WP-install somewhere else, untouched by site-specific stuff.
Note that many badly written plugins (and even a few themes) may make the assumption that they are in the main wp-content folder. Anything doing something stupid like include '../../../wp-load.php'; or similar will be broken by this approach. This error is on the part of the plugin/theme. No plugin/theme should ever include wp-load.