Subtramas sobrepostas empilhadas verticalmente

A principal mudança que você precisa fazer é tornar a cor de fundo Axestransparente e usar um negativo hspacepara forçar os gráficos a se sobreporem um pouco mais:

plt.rcParams['axes.facecolor'] = 'none'                  # transparent Axes background
gs = GridSpec(3, 1, height_ratios=[1, 1, 1], hspace=-.1) # negative hspace for overlap

Ajustar o eixo y no seu segundo gráfico também será necessário aqui, pois atualmente ele corta apenas uma pequena parte dos dados.

ax2.set_ylim(-3.5, 3.5)

Colocando tudo de volta no seu script:

import numpy as np
import matplotlib.pyplot as plt
from matplotlib.gridspec import GridSpec
years = np.linspace(1300, 2000, 700)

np.random.seed(42)

delta_13C = np.cumsum(np.random.normal(0, 0.1, 700))
delta_13C = delta_13C - np.mean(delta_13C)
delta_18O = np.cumsum(np.random.normal(0, 0.08, 700))
delta_18O = delta_18O - np.mean(delta_18O)
temp_anomaly = np.cumsum(np.random.normal(0, 0.03, 700))
temp_anomaly = temp_anomaly - np.mean(temp_anomaly)
temp_anomaly[-100:] += np.linspace(0, 1.5, 100)

plt.style.use('default')
plt.rcParams['font.size'] = 12
plt.rcParams['axes.linewidth'] = 1.5
plt.rcParams['axes.labelsize'] = 14
plt.rcParams['axes.facecolor'] = 'none' # make facecolor transparent

fig = plt.figure(figsize=(10, 8))
gs = GridSpec(3, 1, height_ratios=[1, 1, 1], hspace=-.1) # negative hspace for overlap

ax1 = fig.add_subplot(gs[0])
ax1.plot(years, delta_13C, color='green', linewidth=1.0)
ax1.set_ylabel('First', color='green', labelpad=10)
ax1.tick_params(axis='y', colors='green')
ax1.set_xlim(1300, 2000)
ax1.set_ylim(-4, 4)
ax1.xaxis.set_visible(False)
ax1.spines['top'].set_visible(False)
ax1.spines['bottom'].set_visible(False)
ax1.spines['right'].set_visible(False)
ax1.spines['left'].set_color('green')

ax2 = fig.add_subplot(gs[1])
ax2.plot(years, delta_18O, color='blue', linewidth=1.0)
ax2.yaxis.tick_right()
ax2.yaxis.set_label_position("right")
ax2.set_ylabel('Second', color='blue', labelpad=10)
ax2.tick_params(axis='y', colors='blue')
ax2.set_xlim(1300, 2000)
ax2.set_ylim(-3.5, 3.5) # changed the y-limits ever slightly since the previous clipped data
ax2.xaxis.set_visible(False)
ax2.spines['top'].set_visible(False)
ax2.spines['bottom'].set_visible(False)
ax2.spines['left'].set_visible(False)
ax2.spines['right'].set_color('blue')

ax3 = fig.add_subplot(gs[2])
ax3.plot(years, temp_anomaly, color='gray', linewidth=1.0)
ax3.set_ylabel('Third', color='black', labelpad=10)
ax3.set_xlim(1300, 2000)
ax3.set_ylim(-1.0, 1.5)
ax3.set_xlabel('Year (CE)')
ax3.spines['top'].set_visible(False)
ax3.spines['right'].set_visible(False)

plt.show()

É claro que um toque de estrutura também pode limpar um pouco o roteiro:

from collections import namedtuple

import numpy as np
import matplotlib.pyplot as plt
from matplotlib.gridspec import GridSpec
years = np.linspace(1300, 2000, 700)

np.random.seed(42)

delta_13C = np.cumsum(np.random.normal(0, 0.1, 700))
delta_13C = delta_13C - np.mean(delta_13C)
delta_18O = np.cumsum(np.random.normal(0, 0.08, 700))
delta_18O = delta_18O - np.mean(delta_18O)
temp_anomaly = np.cumsum(np.random.normal(0, 0.03, 700))
temp_anomaly = temp_anomaly - np.mean(temp_anomaly)
temp_anomaly[-100:] += np.linspace(0, 1.5, 100)

plt.style.use('default')
plt.rc('font', size=12)
plt.rc('axes', linewidth=1.5, labelsize=14, facecolor='none')

PlotSettings = namedtuple('PlotSettings', ['data', 'linecolor', 'labelcolor', 'ylabel', 'yrange'])

configurations = [
    PlotSettings(delta_13C,    linecolor='green', labelcolor='green', ylabel='First',  yrange=(-4  , 4  )),
    PlotSettings(delta_18O,    linecolor='blue',  labelcolor='blue',  ylabel='Second', yrange=(-3.5, 3.5)),
    PlotSettings(temp_anomaly, linecolor='gray',  labelcolor='black', ylabel='Third',  yrange=(-1  , 1.5)),
]

fig, axes = plt.subplots(
    nrows=len(configurations), ncols=1, figsize=(10, 8),
    sharex=True, sharey=False,
    gridspec_kw={'hspace': -.1},
)

for i, (config, ax) in enumerate(zip(configurations, axes.flat)):
    ax.plot(years, config.data, color=config.linecolor, linewidth=1.0)

    # Format the X/Y Axes
    ax.set_ylabel(config.ylabel, color=config.labelcolor, labelpad=10)
    ax.tick_params(axis='y', colors=config.labelcolor)
    ax.set_ylim(*config.yrange)
    ax.xaxis.set_visible(False)

    # Format the spines
    ax.spines[['top', 'bottom']].set_visible(False)
    if (i % 2) == 0:
        ax.spines['right'].set_visible(False)
        ax.spines['left' ].set_visible(True)
        ax.spines['left' ].set_color(config.labelcolor)
    else:
        ax.spines['right'].set_visible(True)
        ax.spines['left' ].set_visible(False)
        ax.spines['right'].set_color(config.labelcolor)
        ax.yaxis.tick_right()
        ax.yaxis.set_label_position("right")

# Make special adjustments to the bottom-most plot
axes.flat[-1].spines['bottom'].set_visible(True)
axes.flat[-1].xaxis.set_visible(True)
axes.flat[-1].set_xlabel('Year (CE)')
axes.flat[-1].set_xlim(1300, 2000)

plt.show()