How to preserve axis aspect ratio with tight_layout

I have a plot with both a colorbar and a legend. I want to place the legend outside of the plot to the right of the colorbar. To accomplish this, I use bbox_to_anchor argument, but this causes the legend to get cut off:

import matplotlib.pyplot as plt
import numpy as np
from scipy.stats import norm

_, ax = plt.subplots()

extent = np.r_[0, 1, 0, 1]
space = np.linspace(0, 1)
probs = np.array([[norm.cdf(x + y) for x in space] for y in space])
colormap = ax.imshow(probs, aspect="auto", origin="lower", extent=extent, alpha=0.5)
colorbar = plt.colorbar(colormap, ax=ax)
colorbar.set_label(f"Probability")
ax.scatter(
    [0.2, 0.4, 0.6], [0.8, 0.6, 0.4], color="r", label="Labeled Points",
)
plt.legend(loc="center left", bbox_to_anchor=(1.3, 0.5))
plt.title
plt.show()

Plot with legend cut off

To fix the legend, I insert a call to plt.tight_layout() before plt.show(), but this causes the aspect ratio to get distorted:

Plot with distorted aspect ratio

How can I show the entire legend and preserve the aspect ratio of the axes?

You can manage the ratio between axis height and width with matplotlib.axes.Axes.set_aspect. Since you want them to be equal:

ax.set_aspect(1)

Then you can use matplotlib.pyplot.tight_layout to fit the legend within the figure.
If you want to adjust margins too, you can use matplotlib.pyplot.subplots_adjust.

Complete Code

import matplotlib.pyplot as plt
import numpy as np
from scipy.stats import norm

_, ax = plt.subplots()

extent = np.r_[0, 1, 0, 1]
space = np.linspace(0, 1)
probs = np.array([[norm.cdf(x + y) for x in space] for y in space])
colormap = ax.imshow(probs, aspect="auto", origin="lower", extent=extent, alpha=0.5)
colorbar = plt.colorbar(colormap, ax=ax)
colorbar.set_label(f"Probability")
ax.scatter([0.2, 0.4, 0.6], [0.8, 0.6, 0.4], color="r", label="Labeled Points",)
plt.legend(loc="center left", bbox_to_anchor=(1.3, 0.5))

ax.set_aspect(1)
plt.tight_layout()
plt.subplots_adjust(left = 0.1)

plt.show()