This question is often answered with a resounding No!, but lets look at some numbers.
When charging LFP cells there are generally two important voltages, 3.65V and 3.70V. 3.65V marks the change from constant current to constant voltage, and 3.70V is the absolute maximum charge voltage.
In a 4S configuration, 3.65V per cell is 14.6V, and 3.70V per cell is 14.8V.
Now let’s look at a 12V solar charge controller’s numbers.
14.8V – equalisation.
14.5V – bulk
14.2V – acceptance
13.8V – float
The first thing to note is a 12V LA controller will not exceed an LFP’s maximum voltage.
The next is that the bulk voltage is slightly below an LFP cell’s 3.65V CC/CV change voltage. While this is not bad in the sense of not exceeding the LFP’s spec’s, it may mean that the LFP will not be properly charged. And the same goes for 14.2V and 13.8V.
So, on the face of it, you can connect LFP’s to a lead-acid solar charge controller. It may not be efficient and if may not charge the LFP’s to their full potential, but it can be done.