In my possession is a 1994 RX-7 owner's manual with very stern warnings against using synthetic oil. I don't know if the new RX-8 RENESIS rotary, which is a heavily modified 13B, has the same warnings.
Mobil 1, Amsoil, and Red Line have all been used successfully with racing rotaries, but street use is a different story. You probably know that oil is metered into the fuel to lube the rotor apex seals.
As I understand it, Mazda's concern with using synthetics in rotaries is that synthetic oils do not combust as conventional oils do, so ash builds up, causing detonation. The detergent and additive package in street oils is also an issue: racing oils don't typically use detergents, which also create ash when burned. Detonation will kill a rotary even quicker than it will a piston engine.
Mazda went to the trouble of creating a special synthetic oil for its racing rotaries, including the 787B racer with the 26B four-rotor that won Le Mans in 1991. The chemistry of this oil is different from that of other automotive synthetics and is covered in SAE paper 922375, "The development of lubricating oils for rotary racing engines". (Unfortunately, as I key this, I don't have the paper handy or else I would have mentioned its base.) You can order a copy from SAE; its website is easy to find, and the organization doesn't mind dealing with the public. You can buy a street version of this special oil from Mazdaspeed at roughly $35 a liter (about a quart).
Mazda did not want to use an existing synthetic oil for the grueling 24-hour Le Mans race, during which the car covered several thousand miles at speeds up to over 200 mph. There has to be a reason.
If you use a synthetic oil in a rotary, be careful. I do wonder about one thing. Synthetics are far more common in Europe than here. What oil does Mazda recommend for its RX-7s and RX-8 there?