How to round connection for Matplotlib axis spines

An idea is to hide the spines, and draw a FancyBboxPatch at the same spot. Setting clip_on=False avoids that the box would be clipped inappropriately.

Assigning the rounded box to the ax_a.patch makes the clipping both of the main ax and of the objects inside the inset ax work.

Here is example code modifying the left inset axis.

import matplotlib.pyplot as plt
from matplotlib.patches import FancyBboxPatch
import numpy as np

fig, ax_e = plt.subplots(figsize=(10, 5))

ax_a = ax_e.inset_axes([.1, .1, .1, .2])
ax_b = ax_e.inset_axes([.3, .1, .1, .2])

for ax in [ax_a, ax_b]:
    ax.set_yticks([])
    ax.set_xticks([])
for s in ax_a.spines:
    ax_a.spines[s].set_visible(False)
p_bbox = FancyBboxPatch((0, 0), 1, 1,
                        boxstyle="round,pad=-0.0040,rounding_size=0.2",
                        ec="black", fc="white", clip_on=False, lw=1,
                        mutation_aspect=1,
                        transform=ax_a.transAxes)
ax_a.add_patch(p_bbox)
ax_a.patch = p_bbox

t = np.linspace(0, 2 * np.pi, 100)
ax_a.plot(np.sin(3 * t), np.sin(4 * t))  # drawing some curve
ax_a.plot([-2, 2], [-2, 2], 'g', lw=10, alpha=0.3)  # diagonal line to check the clipping in the corners
ax_a.set_xlim(-1.1, 1.1)
ax_a.set_ylim(-1.1, 1.1)

ax_e.set_facecolor('tomato')  # show the clipping around the inset axes

plt.show()