My vote is poor ground - according to the BMW E36 Electrics Manual it is pin number 24 on Connector X16 (the 26 pin white connector on the back of the cluster) described in the manual as "Terminal 31/Ground for instrument cluster electronics". The other end of this wire is grounded at a strip connector designated X10009 which is located just to the right of the centre console under the dash on LHD cars.
Other ground connections to the dash are via pins 3 of connector X17 (the 26 pin blue connector on the back of the cluster) which provides an earth for the indicator and warning lights and pin 21 on the same connector, which provides an earth connection to the ASC control on later cars. I would say that the earth is loose at X10009 or you have a dry joint inside the cluster as you mention that the connectors are clean and tight on the back of the cluster.
The lights stay on because they are drawing earth from Pin 3 of connector X17.
According to the manual, the wire should be Brown, but as this is my interpretation of the electrics manual for a 98 model E36, proceed with caution.