This sounds like mortar torching. It used to be applied to the underside of tiling to prevent excessive wind and snow blowing into the roof. It could be clay but is normally a lime : sand mortar. This will weaken overtime and fall off.
The roofing cmpany are right in as much as the only way to correctly place an underlay under the tiles and over the battens is to remove the tiling. This raises the issue of ventilation which is currently naturally occuring through the tiles. If you progress down this route, fit a Tyvec or similar breathable membrane to ensure that good levels of ventilation are maintained. This willhelp to reduce nasties such as condensation whic can lead to timber decay and wood worm etc.
Does the roof covering look to be in good condition. Get some binoclears and have a good look from outside (across the road may be). If the tiling looks to be in good condition you may not need an underlay. Underlays are secondary water proofing measures if ou like should the tiling become defective so if the roof covering is in good ocndition you may need not worry. If there is any penetrating damp (particularly to valleys) you'll need to sort this asap.
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