Let $U \subset X$ be affine open. Looking at Topologies, Definitions \ref{topologies-definition-ph-covering} and \ref{topologies-definition-V-covering} it suffices to show that the base change $\{X_i \times_X U \to U\}$ can be refined by a standard ph covering if and only if it can be refined by a standard V covering. Thus we may assume $X$ is affine and we have to show $\{X_i \to X\}$ can be refined by a standard ph covering if and only if it can be refined by a standard V covering. Since a standard ph covering is a standard V covering, see Topologies, Lemma \ref{topologies-lemma-standard-ph-standard-V} it suffices to prove the other implication. \medskip\noindent Assume $X$ is affine and assume $\{f_i : X_i \to X\}_{i \in I}$ can be refined by a standard V covering $\{g_j : Y_j \to X\}_{j = 1, \ldots, m}$. For each $j$ choose an $i_j$ and a morphism $h_j : Y_j \to X_{i_j}$ such that $g_j = f_{i_j} \circ h_j$. Since $Y_j$ is affine hence quasi-compact, for each $j$ we can find finitely many affine opens $U_{j, k} \subset X_{i_j}$ such that $\Im(h_j) \subset \bigcup U_{j, k}$. Then $\{U_{j, k} \to X\}_{j, k}$ refines $\{X_i \to X\}$ and is a standard V covering (as it is a finite family of morphisms of affines and it inherits the lifting property for valuation rings from the corresponding property of $\{Y_j \to X\}$). Thus we reduce to the case discussed in the next paragraph. \medskip\noindent Assume $\{f_i : X_i \to X\}_{i = 1, \ldots, n}$ is a standard V covering with $f_i$ of finite presentation. We have to show that $\{X_i \to X\}$ can be refined by a standard ph covering. Choose a generic flatness stratification $$ X = S \supset S_0 \supset S_1 \supset \ldots \supset S_t = \emptyset $$ as in Lemma \ref{lemma-generic-flatness-stratification} for the finitely presented morphism $$ \coprod\nolimits_{i = 1, \ldots, n} f_i : \coprod\nolimits_{i = 1, \ldots, n} X_i \longrightarrow X $$ of affines. We are going to use all the properties of the stratification without further mention. By construction the base change of each $f_i$ to $U_k = S_k \setminus S_{k + 1}$ is flat. Denote $Y_k$ the scheme theoretic closure of $U_k$ in $S_k$. Since $U_k \to S_k$ is a quasi-compact open immersion (see Properties, Lemma \ref{properties-lemma-quasi-coherent-finite-type-ideals}), we see that $U_k \subset Y_k$ is a quasi-compact dense (and scheme theoretically dense) open immersion, see Morphisms, Lemma \ref{morphisms-lemma-quasi-compact-scheme-theoretic-image}. The morphism $\coprod_{k = 0, \ldots, t - 1} Y_k \to X$ is finite surjective, hence $\{Y_k \to X\}$ is a standard ph covering and hence a standard V covering (see above). By the transitivity property of standard V coverings (Topologies, Lemma \ref{topologies-lemma-composition-standard-V}) it suffices to show that the pullback of the covering $\{X_i \to X\}$ to each $Y_k$ can be refined by a standard V covering. This reduces us to the case described in the next paragraph. \medskip\noindent Assume $\{f_i : X_i \to X\}_{i = 1, \ldots, n}$ is a standard V covering with $f_i$ of finite presentation and there is a dense quasi-compact open $U \subset X$ such that $X_i \times_X U \to U$ is flat. By Theorem \ref{theorem-flatten-module} there is a $U$-admissible blowup $X' \to X$ such that the strict transform $f'_i : X'_i \to X'$ of $f_i$ is flat. Observe that the projective (hence closed) morphism $X' \to X$ is surjective as $U \subset X$ is dense and as $U$ is identified with an open of $X'$. After replacing $X'$ by a further $U$-admissible blowup if necessary, we may also assume $U \subset X'$ is scheme theoretically dense (see Remark \ref{remark-successive-blowups}). Hence for every point $x \in X'$ there is a valuation ring $V$ and a morphism $g : \Spec(V) \to X'$ such that the generic point of $\Spec(V)$ maps into $U$ and the closed point of $\Spec(V)$ maps to $x$, see Morphisms, Lemma \ref{morphisms-lemma-reach-points-scheme-theoretic-image}. Since $\{X_i \to X\}$ is a standard V covering, we can choose an extension of valuation rings $V \subset W$, an index $i$, and a morphism $\Spec(W) \to X_i$ such that the diagram $$ \xymatrix{ \Spec(W) \ar[d] \ar[rr] & & X_i \ar[d] \\ \Spec(V) \ar[r] & X' \ar[r] & X } $$ is commutative. Since $X'_i \subset X' \times_X X_i$ is a closed subscheme containing the open $U \times_X X_i$, since $\Spec(W)$ is an integral scheme, and since the induced morphism $h : \Spec(W) \to X' \times_X X_i$ maps the generic point of $\Spec(W)$ into $U \times_X X_i$, we conclude that $h$ factors through the closed subscheme $X'_i \subset X' \times_X X_i$. We conclude that $\{f'_i : X'_i \to X'\}$ is a V covering. In particular, $\coprod f'_i$ is surjective. In particular $\{X'_i \to X'\}$ is an fppf covering. Since an fppf covering is a ph covering (More on Morphisms, Lemma \ref{more-morphisms-lemma-fppf-ph}), we can find a standard ph covering $\{Y_j \to X'\}$ refining $\{X'_i \to X\}$. Say this covering is given by a proper surjective morphism $Y \to X'$ and a finite affine open covering $Y = \bigcup Y_j$. Then the composition $Y \to X$ is proper surjective and we conclude that $\{Y_j \to X\}$ is a standard ph covering. This finishes the proof.