Some of my beds use a "sand mulch" as a top cover...the rhizome sits in the sand, the roots extend beneath it, into "real dirt".....the sand does slow down some weed germination
, one bed (for AB's ) has a "pea grave" top coat....again, I think it helps ...a bit
. If you are really "anti-chemical", you can use "corn gluten meal" as a pre emergent …..that's about as "organic" as it gets....and, it's what *Preen (for veggie gardens (organic *Prreen)) really is, anyway
. The "regular" *preen (for flowers, etc.) has a "chemical" added to the corn gluten meal, for better control. If you use either....follow the directions ! Apply to the rates given, water it in, and ….STAY OUT OF THE BED ! Wlking in, for pulling weeds from, the bed, breaks the "chemical barrier , and compacts the soil...and then the weeds will begin to germinate
. Even if you don't use anything but a hoe to weed (you are really making a "dust mulch" so the seeds won't germinate as easily) walking in the bed....or a rain....compacts the soil surface, and makes germination easier.
You are correct about the Mulch / Rotting thing....IF the mulch covers , or touches the rhizome (which tends to impede airflow, and hold moisture), but the pathways, or even areas BETWEEN the individual plants could be "mulched"... that might cut down some of the problem, as you would have "less ground to weed"