Need advice to fix a strange link-building strategy

[hema29]

I have a bit of a situation that I need some advice for.

I have taken on a site with an interesting set-up regarding their SEO strategy.

The main site is a catalog of sorts, with pages for various products. On the same server, using the same database or an API, are a bunch of other sites (mostly built using the same template, although there are some variations) that are basically directories with links to the product pages on the main site. The links are all one-way, from the directories to the main catalog site. The main site performs OK but has been losing lots of traffic recently. The link directory sites mostly do not appear for 99% of any related searches.

The site owner is adamant that these extra links from the directory sites are good for business, but I think the main site would be better off without all of the junk links pointing to it.

My question is how good are these links for the main site and what should I advise my client?

Are they all directories? And they are same or alike?