diff --git a/src/sage/numerical/interactive_simplex_method.py b/src/sage/numerical/interactive_simplex_method.py
index 2da9fa85057..a4ce0a7a9c5 100644
--- a/src/sage/numerical/interactive_simplex_method.py
+++ b/src/sage/numerical/interactive_simplex_method.py
@@ -3910,8 +3910,36 @@ def __init__(self, A, b, c, objective_value,
         c = copy(c)
         B = vector(basic_variables)
         N = vector(nonbasic_variables)
+        # Issue #29101: vector does not guarantee that the result is freshly allocated
+        # if the input was already a vector
+        if B is basic_variables:
+            B = copy(B)
+        if N is nonbasic_variables:
+            N = copy(N)
         self._AbcvBNz = [A, b, c, objective_value, B, N, polygen(ZZ, objective_name)]
 
+    def __copy__(self):
+        r"""
+        TESTS:
+
+        Test that copies do not share state with the original::
+
+            sage: A = ([1, 1], [3, 1])
+            sage: b = (1000, 1500)
+            sage: c = (10, 5)
+            sage: P = InteractiveLPProblemStandardForm(A, b, c)
+            sage: D = P.initial_dictionary()
+            sage: D_2 = copy(D)
+            sage: D is D_2
+            False
+            sage: D.enter('x1')
+            sage: D.leave('x3')
+            sage: D.update()
+            sage: D_2 == D
+            False
+        """
+        return type(self)(*self._AbcvBNz)
+
     @staticmethod
     def random_element(m, n, bound=5, special_probability=0.2):
         r"""