This system was solved in May 2020, using solveSystem in Macaulay2 v1.15 with an Intel(R) Core(TM) i55250U CPU at 1.60GHz.
There were 32 solutions found in 0.0944 seconds (with a Bezout bound of 32).
Reference: "Computation of all solutions to a system of polynomial equations" by M. Kojima and S. Mizuno (pages 131157), "Finding all solutions to a system of polynomial equations" by A.H. Wright (pages 125133), and "A simple homotopy method for determining all isolated solutions to polynomial systems" by W. Zulehner (pages 167177).
See also: http://homepages.math.uic.edu/~jan/Demo/wright.html.

The object wright is a method function.