Let

• H = Hét(–, ℚp) : Var/ℚp → Vecℚp
• HdR = Hdr(–/ℚp) : Var/ℚp → Vecℚp

be the two Weil co-homology theories on smooth projective ℚp–varieties that you fixed. Put

Perp := Isom⊗(H, HdR)

(the affine ℚp-scheme whose R-valued points are the tensor-compatible, graded, cup–product–, orientation– and cycle–class-preserving natural isomorphisms
H⊗ℚpR ≃ HdR⊗ℚpR).

1. What is already known?

a) Fontaine–Faltings (and, in the generality you want, Tsuji) give a canonical comparison isomorphism

       ιdR : H⊗ℚpBdR  ≃  HdR⊗ℚpBdR                       (1)

  which is natural in X and compatible with all the extra structure; hence
  ιdR is a BdR–point of Perp.

b) For varieties with good (resp. semistable) reduction, (1) in fact lands in Bcris (resp. Bst); that gives BdR–points of Perp which, after restricting the source category, even live in Spec Bcris or Spec Bst.

c) Because (1) is Gℚp–equivariant, composing it with g ∈ Gℚp produces a new point g·ιdR . In Tannakian language these are exactly the points whose value ring is BdR and which factor through the fibre functor

       H  –––>  Repℚp(Gℚp ).

  Consequently they all lie in the closed subscheme

       Perp(Gℚp) := Isom⊗(ωét, ωdR×BdR)     ⊂ Perp,

  whose Tannaka group is the Zariski closure Ḡℚp of Gℚp in the (conjectural)
  motivic Galois group.

d) Apart from the obvious ambiguity ±1 coming from the orientation constraint (and, on sub–categories, the Bcris / Bst refinements mentioned in (b)), the comparison isomorphism (1) is unique; see e.g. §4 of Faltings, “Crystalline co-homology and p-adic Galois representations”.

In short: every explicit point we know today is contained in Perp(Gℚp).

2. Can we write down a point that is not in Perp(Gℚp)?

   Not with the techniques that currently exist. Producing such a point would amount to constructing a tensor automorphism of the de Rham functor which is not induced by an element of Gℚp. In Tannakian terms that would give a homomorphism

      ℚp → Ḡmot / Ḡℚp
   

   that is not constant, hence would show that Ḡℚp is strictly smaller than the full motivic Galois group; equivalently it would exhibit algebraic cycles that are not detected by p-adic Galois representations, i.e. it would settle new cases of the Tate conjecture over ℚp. No such examples are known, and producing one is expected to be at least as hard as proving the Tate conjecture itself.

3. Summary.

   • The canonical BdR–comparison gives one point, and letting the absolute Galois group act on it gives a whole BdR–orbit; • for good / semistable reduction that orbit even lives inside Bcris / Bst; • apart from these, no other points of Perp have been written down; • constructing a point outside that Gℚp–orbit would yield brand–new information about algebraic cycles over ℚp and is far beyond present knowledge.

So the present state of affairs is that the only explicitly describable points of the “p-adic period torsor’’ are the ones coming from p-adic Hodge theory itself; nothing outside the closed subscheme cut out by Gℚp is known.