We haven't seen the layout: poor construction can cause oscillation. If it's built on a breadboard, with the TLV3501 on one of those SMD to DIP adaptor boards of course, it wouldn't be surprising if it oscillates.
As far as multiplying the frequency by 11 is concerned: harmonic filtering is a method to multiply one frequency by a fixed amount, but it only works over a limited frequency range. It's not impossible that there's a tuned circuit at around the 110MHz mark which is selecting the 11th harmonic, in which case increasing/decreading the input frequency a little will cause the output frequency to change proportionally.
Not impossible, but most unlikely.
A "one step" frequency multiplication of 11 times is very far from the standard frequency multiplication schemes that were used in the past, which mainly used cascaded frequency doublers & triplers, with the
very occasional quadrupler.
The problem with "one step" frequency multiplication by large multiples (& why it was rejected as a viable method), is that the amount of harmonic energy available to provide the multipled output is minimal at the higher harmonics.
If it was found unsatisfactory, even with dedicated high "Q" LC resonant circuits, it makes the possibility of it happening "accidentally", unlikely in the extreme.
Another possibility arises, that the comparator is in fact, oscillating at around 110MHz, but it has become "locked" to the incoming 10MHz signal.
To my mind, it would be more likely to lock to 100MHz, though.
The possibility of oscillation is why I suggested the OP leave all the input connections in place, & just turn off the 10MHz source,then check what appears at the output.
Even if a signal around 110MHz appears at the output, the possibity of "locking" as mentioned sbove still remains
The best way to check if the input & output signals are in sync would be to monitor the 10MHz signal on one channel of an Oscilloscope, triggering from that input, & look at the 110MHz output on the other channel.
If the latter is stationary on the display, it is, somehow, related to the input 10MHz signal, if not, it is just an unrelated oscillation.