Useful to run the following on both local and remote, to get a sense of what's happening:

bridge monitor

VM, where eth1 is connected to the network I want to make my RPi appear in:

ip link add br0 up type bridge             # Create a bridge
ip address add 192.168.167.134/24 dev br0  # Copy the address and netmask of eth1
ip address flush dev eth1                  # Remove eth1's addresses completely
ip link set eth1 master br0                # Attach eth1 to the bridge

Raspberry Pi:

sudo ip link add br0 up type bridge
sudo ssh \
  -o PermitLocalCommand=yes \
  -o "LocalCommand=ip link set tap5 up master br0" \
  -o Tunnel=ethernet \
  -w 5:5 \
  -t \
  root@192.168.167.237 \
  "ip link set tap5 up master br0"