roundabouts need to have the right of way and no traffic signals for the complete circle. I would tag it with junction=circular which should work with your routing software. Please, read the wiki about junctions and the key junction=*.
I will probably face similar situation where a new tram rail is built right through a “real” roundabout. Do not know what the right of way will be and I guess there will be no traffic lights and for sure no barrier. If the tram will have the right of way, I’ll have to change the junction=* to circular.
I had a look at a couple of our local ‘roundabouts’, long signalised & one is junction=roundabout, the other is junction=circular.
The UK has talked about gyratories in the past (there is an infamous one at Hanger Lane in W London), which seems to cover this type of junction: topolgically like a roundabout with all entry roads having give ways, but with some traffic signals to control flow. In the future we might wish to make split circular junctions into gyratories and other circular junctions (US traffic circles are I think the prime use case of circular).