The loosely synchronized clocks are not required for correctness (i.e. to enforce strict serializability), only for performance, i.e. for achieving 1 WAN round-trip execution for conflicting transactions whose execution overlaps (all other transactions execute on 1 WAN round-trip anyway).

The system maintains a logical clock, and the logical clock enforces the strict serializability condition globally. The logical clock is just cheaper to maintain if the coordinator proposes a value that is already correctly ordered by the loosely synchronized clocks.

edit: sorry, I misread your post as interpreting the loosely synchronized clocks as being necessary for correctness. I answer (very loosely) how strict serializability is maintained in a sibling comment, but in brief a dependency set of all conflicting transactions that might potentially execute earlier is maintained, and a transaction executes in logical timestamp order wrt these dependencies (and, hence, also their dependencies). If a transaction and its dependencies never interacts with another transaction or its dependencies, then their execution order is commutative, i.e. any ordering is indistinguishable from any other.