Why does defining MULTISITE true change the database connection in use

I have taken a copy of a multisite install to work on it locally. I've noticed that if I set define('MULTISITE', true); in wp-config, I get the standard database connection error. If I disable multisite, the theme loads but there are plugins missing.

Can anyone suggest what I have missed? Possibly relevant is that my local dev environment is using NGINX and the original is on Apache. I don't have anything specific to multisite in my nginx config.

If I set both of the following in wp-config:

define('MULTISITE', true);
define('SUBDOMAIN_INSTALL', true);

I get redirected to http://http//local.mydevsite/wp-signup.php?new=local.mysdevsite

I also get redirected there when trying /wp-admin