From some research, beam antennas only seem to come as monoband antennas.
Not true — actually 20/15/10m tri-band beams are very popular. Multi-band yagis come in two basic flavors.
Trapped. Works on the same principle as a trap vertical to alter the effective electrical length of the elements at different frequencies, except there are way more elements to ...
The driven element of a yagi antenna is normally a half wave dipole, and it's true that a dipole antenna is resonant on multiple harmonically related frequencies. An ideal 10 m long dipole for example is resonant on 15 MHz, 30 MHz, 60 MHz, 120 MHz etc.
However the distances between the elements along the boom of a yagi required to provide ...
A resonant 20 meter monoband Yagi will not work on 10 meters.
Have you considered a 3-band 10-15-20 Tri-bander? It is a beam, and definitely not a monobander.
There are also smaller tri-band directional beams, such as the Hexbeam.