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This reminds me of NSolve finds real-valued results in version 9, but not in version 10NSolve finds real-valued results in version 9, but not in version 10, in which you can use any of the Method settings from Methods for NSolveMethods for NSolve

"EndomorphismMatrix"
"CompanionMatrix"
"Legacy"
"Aberth"
"JenkinsTraub"

or even a nonexistent method "Foo":

NSolve[eqs, {x1, x2, x3, x4}, Reals, Method -> "Foo"]
(*
  {{x1 -> 23.2915, x2 -> -28.505, x3 -> 55.2247, x4 -> -0.0852683},
   {x1 -> 46.3516, x2 -> -5.36977, x3 -> 9.01972, x4 -> -0.0198153},
   {x1 -> 56.3754, x2 -> -0.000327403, x3 -> -3.16593, x4 -> -1.46634},
   {x1 -> 54.9, x2 -> 3.2, x3 -> -1.80018*10^-7, x4 -> -6.26944*10^-10},
   {x1 -> 52.6254, x2 -> 0.0121417, x3 -> -2.70055, x4 -> -0.976273},
   {x1 -> 52.2298, x2 -> -0.0000285978, x3 -> 0.928352, x4 -> 2.62754},
   {x1 -> 49.4605, x2 -> 0.00100605, x3 -> 0.511294, x4 -> 2.21332},
   {x1 -> 53.0999, x2 -> 4.30992*10^-13, x3 -> 0.000114669, x4 -> 7.99797}}
*)

This reminds me of NSolve finds real-valued results in version 9, but not in version 10, in which you can use any of the Method settings from Methods for NSolve

"EndomorphismMatrix"
"CompanionMatrix"
"Legacy"
"Aberth"
"JenkinsTraub"

or even a nonexistent method "Foo":

NSolve[eqs, {x1, x2, x3, x4}, Reals, Method -> "Foo"]
(*
  {{x1 -> 23.2915, x2 -> -28.505, x3 -> 55.2247, x4 -> -0.0852683},
   {x1 -> 46.3516, x2 -> -5.36977, x3 -> 9.01972, x4 -> -0.0198153},
   {x1 -> 56.3754, x2 -> -0.000327403, x3 -> -3.16593, x4 -> -1.46634},
   {x1 -> 54.9, x2 -> 3.2, x3 -> -1.80018*10^-7, x4 -> -6.26944*10^-10},
   {x1 -> 52.6254, x2 -> 0.0121417, x3 -> -2.70055, x4 -> -0.976273},
   {x1 -> 52.2298, x2 -> -0.0000285978, x3 -> 0.928352, x4 -> 2.62754},
   {x1 -> 49.4605, x2 -> 0.00100605, x3 -> 0.511294, x4 -> 2.21332},
   {x1 -> 53.0999, x2 -> 4.30992*10^-13, x3 -> 0.000114669, x4 -> 7.99797}}
*)

This reminds me of NSolve finds real-valued results in version 9, but not in version 10, in which you can use any of the Method settings from Methods for NSolve

"EndomorphismMatrix"
"CompanionMatrix"
"Legacy"
"Aberth"
"JenkinsTraub"

or even a nonexistent method "Foo":

NSolve[eqs, {x1, x2, x3, x4}, Reals, Method -> "Foo"]
(*
  {{x1 -> 23.2915, x2 -> -28.505, x3 -> 55.2247, x4 -> -0.0852683},
   {x1 -> 46.3516, x2 -> -5.36977, x3 -> 9.01972, x4 -> -0.0198153},
   {x1 -> 56.3754, x2 -> -0.000327403, x3 -> -3.16593, x4 -> -1.46634},
   {x1 -> 54.9, x2 -> 3.2, x3 -> -1.80018*10^-7, x4 -> -6.26944*10^-10},
   {x1 -> 52.6254, x2 -> 0.0121417, x3 -> -2.70055, x4 -> -0.976273},
   {x1 -> 52.2298, x2 -> -0.0000285978, x3 -> 0.928352, x4 -> 2.62754},
   {x1 -> 49.4605, x2 -> 0.00100605, x3 -> 0.511294, x4 -> 2.21332},
   {x1 -> 53.0999, x2 -> 4.30992*10^-13, x3 -> 0.000114669, x4 -> 7.99797}}
*)
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Michael E2
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This reminds me of NSolve finds real-valued results in version 9, but not in version 10, in which you can use any of the Method settings from Methods for NSolve

"EndomorphismMatrix"
"CompanionMatrix"
"Legacy"
"Aberth"
"JenkinsTraub"

or even a nonexistent method "Foo":

NSolve[eqs, {x1, x2, x3, x4}, Reals, Method -> "Foo"]
(*
  {{x1 -> 23.2915, x2 -> -28.505, x3 -> 55.2247, x4 -> -0.0852683},
   {x1 -> 46.3516, x2 -> -5.36977, x3 -> 9.01972, x4 -> -0.0198153},
   {x1 -> 56.3754, x2 -> -0.000327403, x3 -> -3.16593, x4 -> -1.46634},
   {x1 -> 54.9, x2 -> 3.2, x3 -> -1.80018*10^-7, x4 -> -6.26944*10^-10},
   {x1 -> 52.6254, x2 -> 0.0121417, x3 -> -2.70055, x4 -> -0.976273},
   {x1 -> 52.2298, x2 -> -0.0000285978, x3 -> 0.928352, x4 -> 2.62754},
   {x1 -> 49.4605, x2 -> 0.00100605, x3 -> 0.511294, x4 -> 2.21332},
   {x1 -> 53.0999, x2 -> 4.30992*10^-13, x3 -> 0.000114669, x4 -> 7.99797}}
*)