Managing Fax Machines Sharing A Single Analog Line

I support a small medical practice that recently moved to a new location, where they were given tech plates (hatches) in each room. 3 ports each: network, phone (PBX) and an analog line. They bought a fax machine and everything was OK. The problem began when they bought a new multi-function printer with fax capability, and plugged it into the analog line port- now faxes are going to this fax machine and not the one at the front desk.

I'm guessing there's no way two fax machines can *predictably* share a single fax line- am I wrong, and if so, is there a method and/or product I can but which will let me steer all incoming faxes to a specific fax machine on this shared analog connection?