To accomplish what I was looking for, I used the solution found here, and modified it to fit my needs. Here's what I ended up with:
<Style x:Key="SidebarTabControl" TargetType="TabControl">
<Setter Property="Background" Value="#FFC6D3DE" />
<Setter Property="Padding" Value="0,20,0,20" />
<Setter Property="TabStripPlacement" Value="Left" />
<Setter Property="IsSynchronizedWithCurrentItem" Value="True" />
<Setter Property="Template">
<Setter.Value>
<ControlTemplate TargetType="{x:Type TabControl}">
<Grid KeyboardNavigation.TabNavigation="Local">
<Grid.ColumnDefinitions>
<ColumnDefinition Width="200" MinWidth="150" MaxWidth="400" />
<ColumnDefinition />
</Grid.ColumnDefinitions>
<!-- Background of the sidebar and our clipping bounds -->
<Border Grid.Column="0"
CornerRadius="10,0,0,10"
Background="{TemplateBinding Background}"
Name="mask" />
<!-- Border necessary so that the top tab does not get clipped prematurely -->
<Border Grid.Column="0" Background="Transparent">
<!-- Add opacity mask to clip contents as they're scrolled -->
<Border.OpacityMask>
<VisualBrush Visual="{Binding ElementName=mask}"/>
</Border.OpacityMask>
<ScrollViewer
VerticalScrollBarVisibility="Visible"
HorizontalScrollBarVisibility="Disabled">
<Border Padding="{TemplateBinding Padding}">
<TabPanel
IsItemsHost="True"
KeyboardNavigation.TabIndex="1"
Background="Transparent">
</TabPanel>
</Border>
</ScrollViewer>
</Border>
<ContentPresenter
Grid.Column="1"
Margin="0"
ContentSource="SelectedContent" />
<GridSplitter Grid.Column="0"
HorizontalAlignment="Right"
VerticalAlignment="Stretch"
Background="{StaticResource SplitterBrush}"
ShowsPreview="True"
Width="1" />
</Grid>
</ControlTemplate>
</Setter.Value>
</Setter>
</Style>
Edit: I found the solution to the clipping problem with the first TabItem. Nesting the ScrollView inside a second border and applying the OpacityMask to this border, and not the ScrollView fixed the problem. Additionally, I had to explicitly set Background="Transparent" to the Border where the OpacityMask was being applied in order for the clip to not happen prematurely.