Not on an alliance RTW ticket, which prices the whole itinerary in one cabin. A custom multi-stop build lets you choose cabin leg by leg, so you can buy business on the overnight ultra-long-hauls and economy on the short daytime hops. For many travelers that is what makes business class affordable.
Not on an alliance round-the-world product. Those price the entire itinerary in one cabin class, which means paying the business class rate on every single segment including the 90-minute hop from Bangkok to Singapore.
A custom multi-stop build prices each fare component separately, so you can choose cabin leg by leg. Business from Los Angeles to Tokyo (10 hours), economy Tokyo to Bangkok (6 hours), business Bangkok to London (11 hours), economy London to New York (7 hours).
That is the move that makes business class affordable on a lot of round-the-world trips. You put the money on the overnight and ultra-long-haul legs where it changes how you arrive, and save it on the short daytime segments where it does not.
Alliance RTW business class does exist, but availability in the required fare classes is limited. If mixed cabins matter to you, expect a custom build rather than an off-the-shelf product.