You'll need some more advanced logic to prevent backflow from the output to the input.
Two Hall current sensors does the job there, while we do not need high precision current measurements, while need to switch proper mosfets to bypass body diodes when current is above some threshold value eg. 1A.
So, if current is below 1A whatever current direction is we switch all mosfets off than its body diodes conducts, but losess are small lets say about 1.5W per switch, but from Hall current sensors we know which diodes conducts, so if this current is above 1A, we know which mosfets should be bypassed
At the end of half wave cycle (it will depend on load, so it can be inside half wave period, eg. when capacitor load will move this time to different voltage levels) current will drop again below 1A so we have some dead time when current is crossing set threshold (1A in this example).
Additionaly if mosfets are bypassed we monitor if current didn't changed direction (eg. regerative braking tries to push energy back througth our "not ideal but low power losess bridge" and if voltage will be higher than from bridge input, we need to switch off that bypassed mosfet too, to prevent flow in reverse direction).
Using one Hall sensor - since I know current direction- per half-bridge I know which side (low or high) should be bypassed, so control is very easy based on Hall sensor radiometric output.
Maybe it could be made cheaper without Hall current sensors, but I like to know current levels in my bridge anyway, so the same sensors can be used to monitor current.