This system was solved in May 2020, using solveSystem in Macaulay2 v1.15 with an Intel(R) Core(TM) i5-4258U CPU at 2.40GHz.
There were 64 solutions found in 0.525506 seconds (with a Bezout bound of 144).
Reference:
The PoSSo test suite.
See also: http://homepages.math.uic.edu/~jan/Demo/caprasse.html
i1 : caprasse(QQ)
2 3 2 2 3 2 2
o1 = {y z + 2x*y*t - 2x - z, - x z + 4x*y z + 4x y*t + 2y t + 4x - 10y +
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2 3 2 2
4x*z - 10y*t + 2, 2y*z*t + x*t - x - 2z, - x*z + 4y*z t + 4x*z*t +
------------------------------------------------------------------------
3 2 2
2y*t + 4x*z + 4z - 10y*t - 10t + 2}
o1 : List
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The object caprasse is a method function.