You are bringing up a rational point in this thread? Fool!
Seriesly, of course there was mixing. After Ike left Italy in 1943, General Maitland Wilson was appointed Supreme Commander of the Mediterranean Theatre. General Alexander commanded the 15th Army Group under him, which consisted of the 5th US Army and the 8th BritishArmy.
FDR actually wanted Marshall to command all of the Western Fronts, but Churchill vetoed that. FDR then decided Ike would command Overlord instead of Marshall.
Part, though not all, of the screw-up's in the Battle of the Bulge were because US forces on the north of the bulge, were phyicscally separated from their US commenders on the south/center/middle of the bulge and "had to be" under the control of Montgomery.
Much of the screw-ups in the Caen attacks after D Day and in Holland air attack (A Bridge Too Far" was because US troops and strategic policies were determined by the Brit's reluctance to attack without a specifically prepared/too great a demand to attack WITH too poor an attack planning.