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 ...
Dipoles are resonant on odd harmonics of their fundamental frequency. So a dipole resonant on 20 meters, or 14 MHz, is also resonant on 14×3 = 42 MHz, 14×5 = 70 MHz, and so on.
10 meters is (approximately) an even harmonic of 20 meters, so this does not work. At the even harmonics you get zero reactance in the feedpoint impedance, but the resistance is very ...
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.