diff --git a/src/sage/rings/qqbar.py b/src/sage/rings/qqbar.py
index 996cea78a4f..ef4c026505e 100644
--- a/src/sage/rings/qqbar.py
+++ b/src/sage/rings/qqbar.py
@@ -5741,11 +5741,11 @@ def _floor_ceil(self, method):
             ....:            AA(0), AA(1), AA(-1), AA(1/2), AA(-1/2)]]
             [[0, 1, 1, 0], [-1, 0, -1, 0], [1, 2, 1, 1], [-2, -1, -1, -1],
             [17, 18, 18, 17], [0, 0, 0, 0], [1, 1, 1, 1], [-1, -1, -1, -1],
-            [0, 1, 1, 0], [-1, 0, -1, 0]]
+            [0, 1, 0, 0], [-1, 0, 0, 0]]
             sage: [[z.floor(), z.ceil(), z.trunc()] for z in [two, a*b]] # long time
             [[2, 2, 2], [0, 1, 0]]
             sage: [one_half.round(), (-one_half).round()] # long time
-            [1, -1]
+            [0, 0]
         """
         for i in itertools.count():
             candidate = method(self._value.lower())
@@ -5806,9 +5806,9 @@ def round(self):
             sage: AA(sqrt(2)).round()                                                   # needs sage.symbolic
             1
             sage: AA(1/2).round()
-            1
+            0
             sage: AA(-1/2).round()
-            -1
+            0
         """
         return self._floor_ceil(lambda x: x.round())
 
diff --git a/src/sage/rings/real_mpfi.pyx b/src/sage/rings/real_mpfi.pyx
index 4d8dafa0515..f9fd378d4bf 100644
--- a/src/sage/rings/real_mpfi.pyx
+++ b/src/sage/rings/real_mpfi.pyx
@@ -3123,9 +3123,9 @@ cdef class RealIntervalFieldElement(RingElement):
 
             sage: r = RIF(2.5, 3.5).round()
             sage: r
-            4.?
+            3.?
             sage: r.lower()
-            3.00000000000000
+            2.00000000000000
             sage: r.upper()
             4.00000000000000
         """
@@ -3478,28 +3478,22 @@ cdef class RealIntervalFieldElement(RingElement):
             sage: RIF(-pi).unique_round()                                               # needs sage.symbolic
             -3
             sage: (RIF(4.5).unique_round(), RIF(-4.5).unique_round())
-            (5, -5)
+            (4, -4)
 
        TESTS::
 
             sage: RIF(-1/2, -1/3).unique_round()
-            Traceback (most recent call last):
-            ...
-            ValueError: interval does not have a unique round (nearest integer)
+            0
             sage: RIF(-1/2, 1/3).unique_round()
-            Traceback (most recent call last):
-            ...
-            ValueError: interval does not have a unique round (nearest integer)
+            0
             sage: RIF(-1/3, 1/3).unique_round()
             0
             sage: RIF(-1/2, 0).unique_round()
-            Traceback (most recent call last):
-            ...
-            ValueError: interval does not have a unique round (nearest integer)
+            0
             sage: RIF(1/2).unique_round()
-            1
+            0
             sage: RIF(-1/2).unique_round()
-            -1
+            0
             sage: RIF(0).unique_round()
             0
         """
diff --git a/src/sage/rings/real_mpfr.pyx b/src/sage/rings/real_mpfr.pyx
index 10ff6b8baae..421859afdfc 100644
--- a/src/sage/rings/real_mpfr.pyx
+++ b/src/sage/rings/real_mpfr.pyx
@@ -3033,7 +3033,11 @@ cdef class RealNumber(sage.structure.element.RingElement):
     def round(self):
         """
          Round ``self`` to the nearest representable integer, rounding halfway
-         cases away from zero.
+         cases to even (banker's rounding).
+
+         This matches the behavior of Python's built-in ``round()`` function
+         for floats (Python 3+), and Sage's ``Rational.round()`` with default
+         mode.
 
          .. NOTE::
 
@@ -3044,14 +3048,36 @@ cdef class RealNumber(sage.structure.element.RingElement):
              sage: RR(0.49).round()
              0
              sage: RR(0.5).round()
-             1
+             0
              sage: RR(-0.49).round()
              0
              sage: RR(-0.5).round()
-             -1
+             0
+             sage: RR(1.5).round()
+             2
+             sage: RR(2.5).round()
+             2
+             sage: RR(-1.5).round()
+             -2
+             sage: RR(-2.5).round()
+             -2
+
+         This uses round-to-even (banker's rounding) for tie cases, matching
+         Python 3 behavior and IEEE 754 default rounding::
+
+             sage: [RR(n + 0.5).round() for n in range(-5, 6)]
+             [-4, -4, -2, -2, 0, 0, 2, 2, 4, 4, 6]
+
+         Compare with ``Rational.round()``::
+
+             sage: RR(5/2).round() == QQ(5/2).round()
+             True
+             sage: RR(-7/2).round() == QQ(-7/2).round()
+             True
          """
         cdef RealNumber x = self._new()
-        mpfr_round(x.value, self.value)
+        # Use mpfr_rint with MPFR_RNDN (round to nearest, ties to even)
+        mpfr_rint(x.value, self.value, MPFR_RNDN)
         return x.integer_part()
 
     def floor(self):
diff --git a/src/sage/schemes/elliptic_curves/period_lattice.py b/src/sage/schemes/elliptic_curves/period_lattice.py
index 82d9be5132e..119dc344711 100644
--- a/src/sage/schemes/elliptic_curves/period_lattice.py
+++ b/src/sage/schemes/elliptic_curves/period_lattice.py
@@ -1043,8 +1043,8 @@ def basis_matrix(self, prec=None, normalised=False):
             [ 1.26920930427955 0.000000000000000]
             [0.634604652139777  1.45881661693850]
             sage: L.basis_matrix(normalised=True)
-            [0.634604652139777 -1.45881661693850]
-            [-1.26920930427955 0.000000000000000]
+            [-0.634604652139777  -1.45881661693850]
+            [ -1.26920930427955  0.000000000000000]
 
         ::
 
@@ -1775,11 +1775,11 @@ def elliptic_logarithm(self, P, prec=None, reduce=True):
             sage: L = E.period_lattice()
             sage: P = E.lift_x(3)
             sage: L.elliptic_logarithm(P)
-            -1.97657221097437 - 1.05021415535949*I
+            -3.47306312187149 + 0.446276755537628*I
             sage: L.elliptic_exponential(_)  # abs tol 1e-15
-            (3.00000000000000 + 9.20856947066460e-16*I : -5.59022723358798 - 0.0894418024719718*I : 1.00000000000000)
+            (3.00000000000000 + 9.63818638531819e-16*I : -5.59022723358799 - 0.0894418024719718*I : 1.00000000000000)
             sage: L.elliptic_logarithm(P, prec=100)  # abs tol 1e-15
-            -3.4730631218714889933426781799 + 0.44627675553762761312098773197*I
+            -1.9765722109743694676873864341 - 1.0502141553594919125343040138*I
             sage: L.elliptic_exponential(_)  # abs tol 1e-28
             (3.0000000000000000000000000000 - 1.4773628579202938936348512161e-30*I : -5.5902272335879800026836302686 - 0.089441802471969391005702381090*I : 1.0000000000000000000000000000)